Ml

iiiiiJiiiiiiiiiiiiii

itt?Hithiflll«'rrlirfiiriiiiJl)H

■1

i

ft -iftrihiiiHtiiiiuiiii

IfUfSmffH

ii

HpBpppppHHHH

I

ti!

iiiu

IN MEMORIAM
FLORIAN CAJORl

Digitized by the Internet Archive

in 2008 with funding from

IVIicrosoft Corporation

http://www.archive.org/details/collegealgebraOOmetzrich

COLLEGE ALGEBEA

COLLEGE ALGEBRA

BY

WILLIAM H. METZLER, Ph.D.
EDWARD DRAKE ROE, JR., Ph.D.

Professors of Mathematics in Syracuse University
AND

WARREN G. BULLARD, Ph.D.

Associate Professor of Mathematics in Syracuse University

LONGMANS, GREEN, & CO.

91 AND 93 fifth avenue, NEW YORK
LONDON, BOMBAY, AND CALCUTTA

1908

Copyright, 1908,

BY

LONGMANS, GREEN, & CO.

r i 4-v

INTRODUCTION

The facts of Algebra are of minor importance to the
average individual and the subject should not be studied
with the acquiring of these facts as the principal object
to be attained. Algebra studied for the mere body of facts
which it contains is a waste of time. These facts the student
will of course acquire, but the authors believe they should
come as incidentals to the acquiring of the methods and
principles of the subject. The principal object, therefore,
for both teacher and student to keep in mind is the acqui-
sition, not of the facts, but of the underlying methods and
principles, and we believe that when this is done the facts
will be more intelligently comprehended and better retained.

We have endeavored to develop the topics treated in
as logical and scientific a manner as was consistent with
good pedagogy. The ground required for entrance to the
scientific courses of the leading Colleges and Schools, or
that required in the freshman year by the students in the
course in arts has been covered, and in addition the needs of
more advanced students have been kept in mind. Rather
more ground is covered than is laid down in the require-
ments for the examinations in Advanced Algebra by the
College Entrance Examination Board. In any case no diffi-
culty will be experienced in omitting the extra parts if the
teacher so desires.

The major portion of the book has been used in pamphlet
form for several years with good results by the freshmen at
Syracuse University.

TABLE OF CONTENTS

CHAPTER I
Graphic Representation of a Function

Representation of a point

Distance between two points

Locus of a moving point

Equations of the first degree

Equations of higher degree

Solution of equations by graphic methods

Simultaneous equations of the first degree

Simultaneous equations of higher degree

Examples

1
2
3
4

6
10

10, 11

11, 12
2, 6, 8, 9, 10, 11, 12

CHAPTER II

Inequalities

Definitions and notations 13

Theorems 14-17

Examples 17-19

CHAPTER III

Ratio, Proportion, and Variation

Ratio 20

Definitions 20, 21

Theorems 21

Proportion 22

Definitions 22

Theorems 22-27

Variation 28

Definitions 28

Graphic illustrations 28, 29

Theorems 29, 30

Examples 21, 26, 27, 30, 31

vii

Vlll

TABLE OF CONTENTS

CHAPTER IV

Theory of Quadratics

The sum and product of the roots .
Formation of equations with given roots
Factoring quadratic expressions
Solution of quadratics by inspection
Nature of the roots of a quadratic .

Discriminant

Geometric representation of roots .

Every quadratic has two and only two roots

Examples

PAGE

32, 33
33

34

35

36, 37

38

38, 39

41, 42

35, 42, 43

CHAPTER V
Factor, Identity, and Remainder Theorems

Factor theorem ,44

Number of roots of an equation 45

Identity theorem 40

Identity of two polynomials 4G, 47

Remainder theorem 48

Examples 49

CHAPTER VI
Commensurable, Incommensurable, and Imaginary Numbers

Definitions 50, 51, 52

Theory of indices 52-56

Radicals, definitions 57, 58

Reduction of surds 58, 59

Reduction of a mixed to an entire surd ...... 60, 61

Addition and subtraction of surds 62

Multiplication of surds 63-65

Division of surds QQ

Involution and evolution of surds 67

Rationalization ........... 68

Properties of quadratic surds 72

Theorems ............ 72

Square root of a binomial surd 73

Radical equations 74-77

TABLE OF CONTENTS

IX

Complex numbers .....

Definitions

Properties of conjugate complex numbers
Identity theorems for complex numbers .
Graphic representation of complex numbers

. 79
. 80
. 81
. 81
82-91

Examples 56, 57, 59, 60, 61, 62, 63, 64, 05, 67, 69-72, 74, 77, 78, 82, 91-94

CHAPTER VII

Progressions

Arithmetical progression
Definitions
Sum of n terms
Geometrical progression
Definitions
Sum of n terms
Harmonical progression
Examples

. 95

95-97
. 96
. 99
99, 101
. 100
103, 104
96, 97-99, 100-103, 105-107

CHAPTER VIII
Permutations and Combinations

Definitions, notations, fundamental principle .... 108, 109

Permutations 109, 110

Combinations ............ 113

Theorems 113-121

Examples 111-113, 121-123

CHAPTER IX

Binomial Theorem

Proof by induction ....

General term

Some properties of binomial coefficients
Theorems .....
Expansion of a multinomial .
Extraction of roots ....
Examples

124, 125

. 129
. 130
132, 133
134-136
. 136
128, 130, 131, 137, 138

TABLE OF CONTENTS

CHAPTER X
Constants, Variables, and Limits

PAGE

Definitions 139

Theorems ............ 140

Illustrative examples 140-145

Value of

Examples

-y"

u — V

145, 146
. 146

CHAPTER XI

Series

Definitions 147

Theorems 148, 149

Methods for testing the convergency or divergency of a series . 151-160
Examples 160, 161, 162, 163

CHAPTER XII

Undetermined Coefficients

General theory .

Theorems

Development of an algebraic fraction into a series
Binomial theorem for any real exponent
Decomposition of fractions into partial fractions .
General term in the development of an algebraic fraction
Summation of integral series

Examples

Definitions

Theorems

Examples

. 164
164, 165
166-168
169-172
173-181

. 182
183, 184
169, 173, 181, 182, 183, 184

CHAPTER XIII
Continued Fractions

. 185-195

. 185-195

188, 190, 191, 192, 193, 195

CHAPTER XIV

Integral Solutions of Indeterminate Equations of the
First Degree

Particular and general solutions 196, 197

Examples 197, 198

TABLE OF CONTENTS xi

CHAPTER XV
Summation of Series

PAGE

Series whose nth term is in the form

(a+w6)[a+(n + l)6]---[a+(w+m-l)6] 199

Series whose wth term is of the form

L 200

Recurring series 203

Definition and sum of 7i terms 203-205

Generating function 205

Finite differences 206-209

Interpolation 212-215

Examples 201, 202, 206, 209-212, 215, 216

CHAPTER XVI

Logarithms

Definitions 217

Theorems 217-219

Change from one base to another 220, 221

Determination of logarithms of numbers and the use of tables . 221-223

Tables 224, 225

Determination of a number from its logarithm 226

Cologarithms 227

Computation by logarithms, illustrative examples .... 228-231

Exponential function 231-241

Exponential theorem .......... 241

Logarithmic series 241

Calculation of logarithms 242, 243

Examples 220, 230, 231, 243

CHAPTER XVII

Determinants

Definitions and notations 244-247

Theorems 248-252

Minors 253, 254

Expansions 250, 254, 255

Solution of linear equations 256-259

Xll

TABLE OF CONTENTS

Product of two determinants
Examples

PAGE

. 259, 260
247, 248, 263, 261

CHAPTER XVIII
Theory of Equations

Significant term of a polynomial in the case of large and small values 262, 263

/ Development of a function 263, 264

Continuity 265

Theorems 266

Descartes' rule of signs 267

Complex roots enter in pairs 268

Number of real roots between a and h 268-270

Relations between coefficients and roots 270

Cube roots of unity 271

. Symmetric functions ......... 271, 272

!„ Factoring of symmetric and related expressions .... 272-274

Transformation of equations 275-277

Contracted division 277

Geometric interpretation of /'(ic) 280, 281

Rolle's theorem • 282

Multiple roots of /(a:) 283,284

The signs of /(x) and /(a:) on passing through a root oif(x) = 0 . 284, 285

Transformed equation having one term less 285, 286

Solution of the cubic 287-289

Discriminant of the cubic 289

Solution of the biquadratic 290-292

Sturm's theorem 292-295

Solution of numerical equations 296, 297

Horner's method 297-300

Examples . . .264,266,267,271,272,278,279,280,282,300,301

CHAPTER XIX
Miscellaneous Topics

Mathematical induction . . . . . .

Limits

Theorems concerning infinitesimals and infinites .
Theorems on limits ......

Convergency and divergency of some particular series

302-305

. 305

306-309

309-311

. 312

TABLE OF CONTENTS xiii

PAGE

Limits of ratios 313

Theorems on convergence 315

Examples 303, 305, 317

Product of two infinite series 320

Vandermonde's theorem 321

Binomial theorem for any index 321-323

Complex variable as a function of its modulus and argument . . 323

De Moivre's theorem 324

Continuity of f{z) 325

Geometric representation of /(«) 325

Isogonality of /(s) 326-329

Failure of isogonality 329

Fundamental proposition of algebra 330, 331

Index . . « 333

COLLEGE ALGEBRA

CHAPTER I

GRAPHIC REPRESENTATION OF A FUNCTION

o

X'

1. Representation of a Point. If a point is to be located,
it must be clone with reference to some known or fixed
positions, usually with reference to known lines. Thus to
determine the position of a point in a plane, let the two
lines X'X and Y^Y intersect

XT"

at right angles in the point 0.

Then if we are given that a
point P is h units distant from
X' X and a units distant from
Y' F, it is located as one of
four points. The fact that it
is h units distant from X'X
limits it to two lines, one on
either side and parallel to X'X^
and at a distance of b units from
it. Similarly, the fact that it
is a units distant from Y' Y
limits it to two lines parallel to Y' Z", one on either side and
a units distant from it. If these two limitations are imposed
at once, the point P is one of the four intersections of these
four lines.

Definitions. If now we suppose distances measured in the
directions OX and OY affected with the positive sign and

B 1

Fig. 1.

2 COLLEGE ALGEBKA

distances measured in the opposite directions affected with
the negative sign, then tlie point P can be definitely located.

The distances thus measured from OX and OY and
affected with the proper signs are called the ordinate and
abscissa, respectively, or together tlie coordinates of P.

The line X' X is called the axis of abscissas or tlie x-axis ;
and the line Y' Y is called the axis of ordinates or the y-axis.
The point 0 is called the origin. If P is a units to the
right of the ?/-axis and h units above the a;-axis its coordi-
nates are a and h and the point is denoted by (a, 5). If it
is a units to the left of the ^-axis and h units above the
ir-axis, its coordinates are — a and h and the point is denoted
by (—a, 5). The abscissa and ordinate of a point are usu-
ally denoted by x and y respectively. Thus the point x = a^
y = h denotes the point (a, 5), or whose abscissa is a., and
whose ordinate is h ; and the point x= a and y = — h denotes
the point (a, — ^). A point whose coordinates are unknown
is usually denoted by (x^ ?/).

The parts of the plane between OX and OY^ OY and
0X\ OX' and OY' , OY' and OX are called the first,
second, third, and fourth quadrants respectively. The point
P is in the first, second, third, or fourth quadrant, according
as (a-th)^ C^cii ^), ( — «, — ^)? oi' (^^ — ^) a-re its coordinates.

EXAMPLES

1. Locate the points (2, 3), (1, - 2), (- 3, 0), (0, - 1),
(-3, -4), (0,0), (-7,8).

2. To what does x= S limit a point ? To what does
x = — 4: limit a point ? To what does y = 5 limit a point ?
To what does y = — 7 limit a point ?

2. To express the distance between two points in terms of
the coordinates of those points.

GRAPHIC REPKESENTATIOX OF A FUNCTION

Let (a-j, ?/j), (2^2, ^2) ^® ^^^® coordinates of the two points

Pv ^2- Y E,

Then

or

V(:C2-^'l)^ + (j/2-^l)^-*

If P2 is ^^® origin, this becomes

>x

Fig. 2.

3. To find the equation of the locus or path of a point
moving according to some law is to find the equation
satisfied by the coordinates of every point on the locus.
Thus to find the equation of the locus of a point (2:, ?/)
which is always equally distant from the two points (.r-^, y^),
(^2' ^2)' ^^^ have

■VCx - x^y + (?/ - ^i)2 = V(a; - x^y +iy- y^yf,

or 2 {x^ -x^x-\-i {ij^^^ -yi)y + ^1^ - x.^ + yi^ -yi^^-

Where does this locus cut the line joining the two points ?

Again, to find the equation of the locus of a point which
moves so that it always remains at a distance of seven units
from the origin, we have

or a;2 + ?/2=49.

Of what locus is this the equation ?

* Since the distance between two points is positive, we neglect the two-
fold sign before the radical.

COLLEGE ALGEBRA

In general, to find the equation of the locus of a point
which moves so that it always remains at a distance of r
units from a fixed point (a, /3), we have

or {x — of + (y — yS)^ = T^'

This is evidently the equation of the circle whose radius
is r and whose center is at (a, /3).

4. To 'plot a point is to locate the point in the plane by
means of its coordinates, and to plot a curve is to trace it by
means of its points or otherwise.

5. Graphs of Equations of the First Degree. We have
seen that the equation x=% represents a line parallel to the
^-axis and at a distance of three units to the right of the
origin. It will be shown by means of graphical represen-
tation that abstract algebraical relations (equations) between
two unknown quantities can be represented in concrete form
as geometrical curves.

6. Let us consider the equation

y =lx -\- h.

Every pair of values of x and y which satisfy the equa-
tion may be taken as the coordinates of a point in a plane,
and the assemblage of all points whose coordinates are solu-
tions of this equation is called the locus or graph of the
equation.

Let {x^, ^j), (x^, ^/g), (rrg, y^ be the coordinates of any
three points Pj, P^, Pg, on this locus. Then w^e have

y^ = Ix^ + 5, (1)

GRAPHIC REPRESENTATION OF A FUNCTION 5

Subtracting (1) from (2) and also from (3), we have

^2 - ^1 = K^l - ^l)'

^3-^1 = ^(^3-^1)-

Dividing (4) by (0), we have

y^ — y\ — -^2 ^1

Vz - Vi H

— X.

O)
(5)

(6)

>x

Fig. 3.

J^\-ti — X^ Xy,

HP 2. = ^2 - ^1'

^^A = ^3 - ^r

The triangles Py^^H and P^P^T^ are right triangles, and
by equation (6) have their sides including the right angles
proportional, and are therefore similar, so that the angle
KP^P^ = the angle HP^P^, and therefore the points P^, P^^
Pg lie in the same straight line. Since Pj, P^, Pg are any
three points, it follows that the locus is a straight line.
From this it is seen that every equation of the first degree

6 COLLEGE ALGEBRA

of this form represents a straight line ; and the most gen-
eral equation of the first degree Ax-\-B'i/-{- C=0 can be

A 0
written in this form (^B ^ 0), thus y = — —x If JB = 0,

B B

the equation Ax + (7= 0 is a line parallel to the ?/-axis. It fol-
lows that every equation of the first degree represents a straight
line. For this reason equations of the first degree are often
called linear equations.

From the fact that every equation of the first degree
represents a straight line, and since two points determine a
straight line, in plotting an equation of the first degree it is
only necessary to plot two of its points.

Thus, for the equation

2 a; + 3 ?/ = 6
when x = 0, y = %

when y =^^ 2; = 3,

which gives us two points where the line meets the coordi-
nate axes.

7. Plot the lines :

1. ?/ = 2a; + 3.

2. y = x.

3. 2a;-7?/ + 4 = 0.

4. x-[- y = 0.

8. Graphs of Equations of Degree higher than the First.

Consider the equation x^ -\- y"^ = -^^ . We know from 3 that
this is a circle with center at the origin and with radius 7.
From the equation itself we can easily see that the locus is
symmetrical with respect to the axis of x ; for if we give x
any value, there will be two values of y equal but opposite
in sign. Similarly, the locus is seen to be symmetrical with
respect to the ^-axis.

GRAPHIC REPRESENTATION OF A FUNCTION

Again, consider the equation ?/2 = 4 a;, and we see that the
locus is symmetrical with respect to the 2;-axis, but is not
with respect to the ?/-axis.

9. If we have two equations in x and y^ each will repre-
sent a curve, and the common solutions of the two equations,
considered as simultaneous, will represent the coordinates
of the intersections of the two curves.

Thus the circle x^ + y^
= 25 and the line x-{- y ='Z
intersect in the two points
whose y coordinates are
given by the equation

Y

or

(3 - ^)2 ^yl = 25,

3±V4T

y^

The X coordinates of the

. , StVTi

same points are ,

\

'

^

X

\

/

\

\

/

\

\

'

\

\

0

\

\

\

V

\

/

\

\

/

\

^_

-^

X

Y'
Fig. 4.

found by substituting the

values of y in the equation of the line.

10. Expanding the equation of the circle obtained in 3,

we have

x^ + ?/2_ ^2ax-1^y-^ iC- + jS^- - r2 = 0,

which shows that in order that the equation of the second

degree

ax^ -\- 2 hxy + hy^ -\- 2 yx + 2fy -\- c = 0

may represent a circle, we must have a = h, and 7i = 0; then
the equation becomes, after dividing by a,

9 9 2 ^ 2/ e ^ X /
a a a

8 COLLEGE ALGEBRA

Comparing these two equations, we have

a a a

which shows that the coordinates of the center are — - and

f . .

— -, and that the radius is
a

r — - V/^ + ^^ — ^^.
a

Thus, to find the center and radius of the circle

we have « = 3, yS = 4, and r = V9 + 16 — 16 = 3.

11. EXAMPLES

1. Plot the points (- 5, 7), (- 1, - 10), (3, - 5).

Find the distance between the points :

2. (2, 3) and (5, 7).

3. (4, 6) and (2, 3).

4. Obtain the formula for the distance between the points
Pj (^r y\) ^^^ -^2 (^2' y^ when they are situated anywhere
and not both in the first quadrant as in 2.

Thus, for Pj in the second quadrant and P^ in the fourth
quadrant we should have as before.

■2 iK/rrt 2

P^P^ = MP^-\-MF^,
MP^ = MQ+QP^
= -QM-{-QP^,

GRAPHIC REPRESENTATION OF A FUNCTION

9

Fig. 5.

since QM taken in the opposite direction must be regarded
as the negative of MQ,

or MF^ = — x-^-\- x^ = x^ — x^, ( QM= x^, QP^ = x<^.)

Similarly, P^M= P^R + RM

= - RP^ + RM

= -^1 + ^2 = ^2-^1-

Therefore

I\P^ = ix^-x,y + iy^-y{)\

P^P^ = V(x, - x{)'^ + iy^ - y,)\

Find the distance between the points :

5. (2, -4) and (4, 6). 7. (2, 3) and (-7, -10).

6. (2, - 3) and (3, - 2). 8. (- 3, 6) and (2, - 4).

Find the equation of the locus of a point which is equally
distant from :

9. (2, 3) and (5, 7). 10. (2, -5) and (-3, 4).

11. (-3, 6) and (2, -4).

10 COLLEGE ALGEBRA

Find the locus of a point which is always at the distance of:

12. 5 units from the origin.

13. 3 units from the origin.

14. 9 units from the origin.

15. Plot the locus in each of the problems 9-11.

16. What angle does the locus make with tlie lines joining
the two fixed points ?

17. Through what point in the line joining the two fixed
points does the locus pass in each case ?

Plot the lines :

18. 3 a: — y + 5 = 0.

19. 2x-\-^+ b = 0.

20. x-\-S y = 0..

21. Plot the curve y^=S6 — x^.

Find the coordinates of the point of intersection of :

22. The line 2:4-2y— 3 = 0 and the curve y^ = 4: x.

23. The line 12 a: — 5 ?/ = 169 and the circle oP'-\- y^ — 169.

Find the center and radius of each of the circles represented
by the following equations :

24. a;2 + ?/2 _ 4 a: + 10 ?/ - 71 = 0.

25. a:2 + ^2_|_2^ + 8^ + 16 = 0.

26. 2;2^^2_^_l_l()^_^25 = 0.

SOLUTIONS OF EQUATIONS BY GRAPHICAL METHODS

12. Simultaneous Equations of the First Degree. We have
seen that to every equation of the first degree in x and y
there corresponds a straight line ; and that the infinite num-
ber of pairs of values of x and y which satisfy this equation

GRAPHIC REPRESENTATION OF A FUNCTION

11

are the coordinates of the points on the line. If we have
two equations of the first degree in x and y, it is apparent
that the common solution of the two equations is represented
by the point of intersection of the two lines.

A graphical solution of simultaneous equations of the first
degree in x and y consists in plotting the lines and determin-
ing the point of intersection by measurement. It is evident
that the solution thus found
is only approximate and that
the degree of approximation
obtained depends upon the
accuracy of the measure-
ment. This method is often
employed by engineers.

Thus to solve the equa-
tions , -, rs

^ — y + 1 = 0,
a; + 7/ - 5 = 0,

we have, plotting the lines
and measuring, 2:= 2 and j/= 3.

Y

1

\.

/

/

/

\

,^

/

\

vV

\

^\

/

\

^/

\

/

\

fv

/r

\

^

/

1

\

^^

/

\

^n

/

\

X

Y'

Fig. 6.

32; + 4?/-17 = 0.

XI

Solve graphically the following equations :

1. 2:+ 2?/ — 3 = 0, 3.
2a;-?/-l = 0.

2. 4 :?; + 7 ?/ + 25 = 0, 4. 7 re - 3 ?/ + 3 = 0,
32:-2?/ + ll = 0. 4a;-o?/=0.

13. To solve graphically two equations of the form
rj^j^yi^ 2gx + 2fy + c = 0,
and Ax + By + C= 0,

we find the center and radius of the circle represented by
the first equation and draw the circle, then plot the line

12

COLLEGE ALGEBRA

represented by the second equation, and measure the coordi-
nates of the points of intersection.

Thus, if we have the equa-
tions

and a; — ?/ + 7 = 0,

we have

« = -4, /3 = 3,

X

'

k

/

r'<^

/^

*

,y

r

/

"^

/C

/

/

V

/

X

^

/

Fig. 7.

O

and r = Vl6 + 9-21 = 2,

and the figure (7) for the
solution.
X Solve graphically the fol-
lowing equations :

1. a;2 + ?/2-|.10:z:-8i/-8 = 0, a;=j/.

2. a;2+i/2_^12a;-f-14?/-15=0, a:+^-f5 = 0.

14. To solve graphically two equations of the form
a:2 + ?/2 _f. 2^:^: + 2/j/ + c = 0,
and 2:2 + / 4- 2^3; 4. ^fy + c' = 0,

we describe the circles represented by the two equations and
measure the coordinates of the points of intersection.
Solve graphically :

1. a:2 + y2_4^_6^_3 = 0,
2 2:2 + ^2 ^ 25,

3. a:2+z/2- 6?/ = 7,
a:2 + ?/2 — 4 ?/ = 7.

CHAPTER II

INEQUALITIES

15. Definition. Any numher a is said to he greater than any
other number h when a— h is positive ; and any number a is
said to be less than b wheii a— b is negative.

Thus 1 is greater than —5, because 1 — (— 5), or 6, is
positive ; and — 7 is less than — 3, because — 7 — (— 3), or
— 4, is negative.

In accordance with this definition zero must be regarded
as greater than any negative number.

' 16. Definition. In algebra an inequality is defined to he the
statement that two numbers are unequal. From this definition
it follows that one number is greater than the other, and
the statement of inequality often includes the information
as to which is the greater. In the discussion which follows
we shall usually treat inequalities of the latter type.

17. Notation. The symbols >, <, :9b are used as signs of
inequality. In the case of > and < the opening is toward
the greater number. Thus a > 5 is read " a is greater than
5." A stroke through the sign =, >, or < negatives its
significance. As examples of the use of these symbols we

^^^^^ 5>3, 2<6, 2:^3, ^>Q, 4<3.

18. Definition. Tlie members of an inequality are the num-
bers compared.

13

14 COLLEGE ALGEBRA

The inequalities a>h^ c > t? are said to subsist in the
same sense, and the inequalities a > 5, c<^d are said to subsist
in the opposite sense.

19. Theorem. If a>h, and h>c, then a'>c.
Proof. a — h 'd^wd b — c are both positive.
Therefore (a — h')-\-(h — c'), ov a — c is positive and a> c.

20. Theorem. If the same number be added to or sub-
tracted from each member of an inequality^ there results an
inequality subsistijig in the sa7ne seiise.

Proof. Suppose a>b; then by definition a — ^ is posi-
tive ; therefore the numbers a -]- c — (b + c) and a — c —
(^ — c) are both positive, since each is equal to a— b.

Hence a -\- c>b -{- c,

and a — c>b — c.

Corollary. Any term of an inequality may be transposed
from one member to the other by changing its sign.

Thus in the inequality a + 5 > c we may subtract b from
each member and obtain a> c — b.

21. Theorem. If the members of an inequality be iyiter-
changed, the sig7i of inequality must be reversed.

Proof. If a>b, a — b is positive, b — a is negative, and
therefore b<a.

22. Theorem. If the signs of both members of an inequality
be changed, the sign of inequality must be reversed.

Proof. Suppose a>b;

then -h> -a, (20, Cor.)

and therefore — a< — b. (21)

INEQUALITIES 15

23. Theorem. An inequality ivill subsist in the same sense
after each member has been multiplied or divided by the same
positive number.

Pkoof. Suppose a>b ; then by definition a — b is posi-
tive, and therefore, if m be positive, m(^a — b) and —(^a—b^

a b ^

are positive, and hence am > bm and — > — .

m m

24. Theorem. If the reciprocals of both members of an
inequality between positive numbers be taken, the sign of in-
equality must be reversed.

Proof. Suppose a>b. Then dividing both members of
the inequality by ab., we have

i>-, (23)

0 a

and therefore - < -• (21)

a b

25. Theorem. If the members of an inequality be multiplied
or divided by the sa7ne negative 7iumber, the sign of inequality
must be reversed.

Proof. Suppose a>b; then —a<-b. (22)

— ac< — be, and < (23)

c c

26. Theorem. If the corresponding members of inequalities
subsisting in the same sense be added, the sums tvill be unequal
in the same sense.

Proof. Suppose a-^ > b-^, «2 > ^v ^3 > ^3' " "'» ^m > ^m-
Then by definition a^ — b^, a^—b^^ a^ — b^, •••, ajj^—bj^,
are positive.

Therefore ^^ — ^^ + a^ — ^2 + ^3 ~ ^3 + ' " + ^w ~ ^»i ^^ posi-
tive, and rtj +a2 + a3+ ••• -f- a,,,>b^-\-b^-\-b^-\ [-b^.

16 COLLEGE ALGEBRA

27. Theorem. If the corresponding members of inequalities
between positive numbers and subsisting in the same sense be multi-
plied together^ the products will be unequal in the same sense.

Proof. Suppose a-^ > Jj, a^ > b^^ a^>b^, • • •, a,„ > b,^.
Then a^a^ > a^^^ (23)

and «2^i ^ ^1^2*

Therefore a^a^ > b^b^^ (19)

which shows that the theorem holds for any two inequalities.
Therefore it holds for the inequalities

a^a^ > b-J)^^
and ^3 > ^3,

and hence a-^a^a^ > b^b^b^,

and by repeating this process we arrive at the general result

a^a^a^ • • • ^m > ^1^2^3 ' * ' ^tn'

28. It follows from 27 that if a and b are positive and
a>b, then a" > V,

and therefore a~^ < 5~% (24)

where n is any positive integer.

29. The subtraction of corresponding members of two
inequalities subsisting in the same sense does not necessarily
give an inequality subsisting in the same sense. Likewise the
division of the members of one inequality by the corresponding
members of another inequality subsisting in the same sense
does not necessarily give an inequality subsisting in the same
sense. The truth of these statements is readily seen by con-
sidering the inequalities 6 > 4

and 3>1.

Subtracting member from member would give 3 > 3.

Dividing member by member would give 2 > 4.

INEQUALITIES 17

30. Theorem. If two numhers are unequal^ the sum of their
squares is greater than twice their product.

Proof. Let a^b; then (a — by > 0, since the square of
any positive or negative number is always positive.

Therefore a'^ - 2 ab -{- b'- > 0

and «2 + 52 > 2 ab. (20, Cor.)

We shall call this the Fimdartieyital Inequality.

31. The principle involved in the last tlieorem may be
extended to expressions of a degree higher than the second.
Thus we may have the theorem : If two positive numbers are
imequal^ the sum of their cubes is greater than their sum multi-
plied by their product.

Proof. From the result of 30 we have

a^^ab-\-b'^>ab. (20, Cor.)

Therefore {a^ - ah + b'^^^a + ^) or a^^b^> ab(a + b). (23)

By similar processes corresponding theorems may be obtained
for the higher degrees.

32. EXAMPLES

1. Find the limit of x in the inequality ^ x -{- 1 > --\- - -

Multiplying both members of the inequality by 15, (23)
we have 752: + 105 > 5 a: + 12,

or 70 2:>-93, (20)

and ^^~1' ^^^)

2. Find the limits of x when the inequalities
x-^>S-{-~ and 1< ^

4 4 x-d

subsist simultaneously.

18 COLLEGE ALGEBRA

From the first inequality we find x>5 ; hence x— S is posi-
tive^ and therefore, from the second inequality,

1>^, (24)

S>x-S, (23)

6>x, (20)

and therefore 6>x>5.

3. Find the limit of x in the inequality

(62; - 5)(2: + 4)> (3:?; + 2)(2 2; + T).

4. Find the limits of x, given that

(2 + 3 a;) (1 - a;) + 3 > 2 a; - 3 a;2,

and (3 2^ + 1) (2; + 1 ) - 17 2: > (3 2; - 2) (a; - 5) + 10.

In the following problems the letters are supposed to
represent positive and unequal quantities.

5. Which is the g-reater, or — ?

^ 2 a-\-b

6. If «2 _j_ 52 _. ][ ^YLd x"^ -\- y^ = 1, sliow that ax-\-hy < 1.

7. If a? -{- 1)^ -\- c^ = 1 and x^ -\- y"^ -\- z^ = 1^ show that
ax -[-hy + cz<. 1.

8. Show that ah(^a + ^) + hc(h + c) + (?«(6' + a) > 6 ahc.

9. Show that (a + J) (5 + (?)((? + a) > 8 a6(?.

10. Prove a^ + 52 + 6'^ > a6 + ^(? + c^-

11. Show that (^a + h — cy+i^c + a — hy+ih + e — ay> ah
+ be -\- ca.

12. The sum of any positive fraction and its reciprocal is
greater than 2.

13. Show that a2J2 _^ j2^2 _^ ^2^2 -> ^j^^^^ _|. 5 _}_ ^).

14. Prove ^3 4- 2^*3^3 «52.

INEQUALITIES 19

15. Prove a^ + b^ > a% + ah^.

16. Show that 2 (a^ + h"^ + ^3) -^ ah (^a + h^ -{- he (h + c)
-\- ca(^c + a).

17. Show that rt^ + ^^3 _|_ ^3 > 3 «5(?.

18. Show that 2 ^3 + 3 53 > 4 a52 + a^J.

19. Show that (x^y + ^/^^ + zH) (xi/ + ^s;^ + rf) > 9 .-rV^s.

CHAPTER III

RATIO, PROPORTION, AND VARIATION

RATIO

33. Definition. If a and h are two quantities of the same
kind^ the ratio of a to h is the quotient of a divided hy h.

If the ratio of a to 6 is r, then a = hr^ which shows that a
is r times 5.

The quantities a and h may be concrete, but their ratio is
abstract.

34. The ratio of a to ^ will be expressed as - or as a : h.

0

The quantities a and h are called the ter77is of the ratio. The
quantity a is called the antecedent and h the consequent.

35. Definitions. The ratio will be one of greater inequality^

a imit ratio, or one of lesser inequality according as a = b.

a o

If we multiply together the ratios - and -, the resulting

6 d

ft p • •

ratio -— is said to be the ratio compounded of the two ratios
hd

a c a o

- and -, or the compound ratio of - and -•
0 d 0 d

EXAMPLES

1. Find the ratio compounded of | and |.

2. Find the ratio compounded of the compound ratio of ^
and |, and 1^ and |.

20

RATIO, PROPORTION, AND VARIATION 21

36. Definitions. The ratio compounded of the ratio -

2 ^

with itself is — and is called the duplicate ratio of -. The

(1 ft

compound ratio — is called the triiylicate ratio of - •

(T 0

37. Definitions. The ratio is called the subdupUcate

fl CI • • ft

ratio of -, and — is called the ^ubtriplicate ratio of -•

EXAMPLES

1. Find the duplicate ratio of |^f , the subdupUcate ratio of J^.

2. Find the triplicate and subtriplicate ratios of ^-^,

38. Definition. The ratio of 6 : a is called the inverse of the
ratio a\h.

39. Theorem, i?^ a series of finite ratios ivliich are not all
equals hettveen positive nu7nhers, the ratio of the sum of the
antecedents to that of the consequents is less than the greatest,
and greater than the least of these ratios.

Proof. Let the ratios ^, -^, -^, •••,—, be denoted by

Vl V^ Vg v„

^r ^2, T-g, •••, r„, the least of these by r and the greatest by B.
Then ^ = r^,

Uc

'_ /yi

2'

^-2

— = r,

V

whence, by clearing each of these of fractions and adding the
corresponding members of the resulting equations we have
the equality

Wi + 1^2 + ^3 + • • • + ^« = ^1^1 + ^'2^'2 + ^'3^3 + • • • + r„v,,.

22 COLLEGE ALGEBRA

If in place of the right member we form expressions con-
taining R and r instead of r^, r^, •••, r,,, it is obvious that the
resulting expressions are greater and less, respectively, than
the left member of the equation, that is,

whence dividing by

we obtain R > ^ ^ — ^-^ — - > r,

which proves the theorem.

PROPORTION

40. Definition. Four quantities a, J, c, d^ such that the ratio
a : h is equal to the ratio c : d^ are said to he in proportion or
to form a proportion. It is to be observed that the quantities
a and b must be of the same kind and that c and d must be
of the same kind, but are not necessarily of the same kind as
a and h.

41. Definitions. In the proportion a : h = e : d^ a and d are
called the extremes, and b and c are called the means,

THEOREMS IN PROPORTION

42. In any proportion the product of the means is equal to
the product of the extremes.

Let the proportion hQ a i b = c : d,

a c

Clearing of fractions, we have ad=bc.

43. In any proportion the terms are in proportion by alter-
nation, that is, the means or the extremes can be interchanged.

RATIO, PROPORTION, AND VARIATION 23

Given the proportion a : h = c : d,

a c

Multiplying both members by -, we have ~ ~ j*

As^ain, multiply both members by - and we have - = - •

a ha

44. In any proportion the terms are in proportion hy in-
version, that is^ the terms of each ratio can he interchanged.

a c
Given the proportion, i — ~f

(J Cv

whence, - = — ,

a c

h d

h d
or, - = —

a c

45. -Zf four quantities are in proportion^ they are in propor-
tion hy addition, that is^ the sum of the first tivo is to the second
as the swn of the second tivo is to the fourth.

a e

Given the proportion, 7 = -,

0 d

whence,

h d

or,

a-\-h c A- d
h d

Again,

h _ d

a-^h e+ d'

nr

a c

Ui,

a-}-h c -\- d

Ao-ain.

a-\-h c -{- d

(44)

multiplying by | = |

(44)

a c

Let the student state the theorem for each of these forms.

24 COLLEGE ALGEBRA

46. If four quantities are in proportion^ they are in propor-
tion hy subtraction, that is^ the difference of the first two is to
the second as the difference of the second two is to the fourth.

a G
Given the proportion, t = -^^

0 d

whence, - — 1 = - — 1,

0 d

a— h c — d

(If*

Multiplying both members of this by - = -, we get

0 d

a c

a — h c — d

Finally, ^^ = ^^. (44)

a c

Many authors use the terms composition and division for
what we have called addition and subtraction.

47. If four quantities are in proportion^ they are in propor-
tion hy addition and subtraction.

Given the proportion, a : h = c : d,

then, a_±h^^±d ^^^ ^^^-^

h d

a— h c — d

Dividing (1) by (2), we have

b d

a -\-b c -\- d

(2) (46)

a — b c — d

48. The quantities a, ^, c, d, •••, are said to be in continued
proportion ii a:b = b : c = c : d = "•.

RATIO, PROPORTION, AND VARIATION 25

49. If a :h = h : c, then h is called a mean proportional to
a and <?, and c is called a third proportional to a and h.

50. If f = ^ = ^=f, then ^"^^"["^"j"f is equal to any

b d J h b-\- d -f-/ + /i

one of the ratios.

For let « _ ^ _ ^ _ ^ _ ^ .
b d f h '

then a = br,

(1)

c = dr,

(2)

e=fr,

(3)

g = Ar.

(4)

Adding (1), (2), (3), (4), we have

a^-c + e + g = (b + d+f+ h)r,

^ a-\- c -\-e + a a e
whence, , ' , — f = r = - = - = •••.
b + d^f-\-Ji b d

K1 Tf a c e g

, 3/3 a^c + 2 ace + 5 g^ + -I aeg

then \/ ^ .^2^ + 2 ^»cf/ + 5/3 + 4 bfh

is equal to any one of these ratios.

For let « _ £ _ ^ _ .£ = .

b~'d~f~h~^'

then a = 5r, (1)

c = dr, (2)

g =/r, (3)

g = hr. (4)

Squaring (1) and multiplying the result by three times

(2) we have Sa^c=Sb''drK (5)

26 COLLEGE ALGEBRA

Multiplying (1), (2), (3) together and multiplying the result
by 2, we have 2 6«e^ = 2 hdfA (6)

Cubing (3) and multiplying the result by 5, we have

^e^=^fh^^, (7)

Similarly, 4 aeg = 4 bfhr^, (8)

Adding (5), (6), (7), (8), we have

3 A + 2 ac^^^ + 5 ^3 + 4 a^^ = (3 hH + 2 J^f + 5/^ + 4 5/A)r3.

,^,, 3 A + 2 a^e + 5 e^ + 4 a6?(7 o
Whence Tmr, t^-t^tt. — i^t^ rr:^ = ^ ?

3ra + 266?/+5/3 + 4/>/A

3 a% + 2 ac£? + 5 e^ + 4 (^ei^ _ _ci _ c _
^^' ^ 3 526? + 2 5(^/ + 5/3 + 4 ^//i "" ^ ~ 5 ~ ^ ~ " * '

From the mode of obtaining this result it is apparent that
a much more general result might be obtained on observing
that the numerator of the fraction is homogeneous in the
antecedents and the denominator is homogeneous in the con-
sequents and of the same degree as the numerator and that
the degree of either is the same as the index of the root. It
is also to be observed that coefficients of corresponding terms
in the numerator and denominator may be any numbers as
long as they are the same.

52. EXAMPLES

1. What is the duplicate ratio of 3 : 4 ? the subduplicate
ratio of 36 : 25 ? the subtriplicate ratio of 1728 : 27 ?

2. Two numbers are in the ratio of 2 to 3, and if 9 be added
to each they are in the ratio of 3 to 4. Find the numbers.

3. Show that the ratio a:h \^ the duplicate of the ratio

a-\- c :h -\- e if e^ = ah.

RATIO, PROPORTION AND VARIATION 27

^^ a c e , ,, , ma? -{- iiac + ve^ a?

4. If 7 = ^ = :^' show that —H)-, — ,-, , /H2 = T5'

5. A ratio of greater inequality is increased and one of
lesser inequality is diminished by taking from both terms
of the ratio any quantity which is less than each of them,
all the numbers involved being positive.

Q If -^ "^^ = — , find X without clearing of fractions {i.e.
1 — X n
without cross multiplication).

7. If -^ = -1 = -2 = -3 = T'^ prove that

^0 ^1 h h

K + 3 Kh + 3 hK + 3 h^h} + h/

8. If 7 = -^, and a, 5, c, d are positive and in order of

0 d

magnitude, prove that a-^ d>h -\- c.

9. If ax^ + 2 5a;?/ + cy^ = 0, find the ratio of x to ?/.

10. The number of students in Syracuse University in 1905
was 2451; in 1906 the number was 2776. What is the ratio
of the increase to the number in 1905 ? (This ratio is called
the rate of increase).

11. If a — h:c — d = a-\-h:c + d prove that a:h = c:d.

12. li a-\-h:c -\- d= ah : cd prove that ac : bd = c — a:h — d.

13. If a:b = c:d, prove ab -^ cd is a mean proportional

between a'^-\-c^ and b^-i-d^.

J, J -, a^ + c^ ja + cY

14. It a\ b = c: d, prove —. ^ = —r-^ — j-^-

^ ¥ + d^ {b + df

15. If x — y:X=y — z: Y=z — x: Z, where x, y, z are
unequal, then X+ Y-\- Z=0.

16. U a(y + z) = b{z + x~) = c(x-hy},
then

x—y _ y — 'z _ z — x

c (a — 5) a (b — c) b{c— a)

28

COLLEGE ALGEBRA

VARIATION

53. Definition. If the ratio of two quantities x and y is
constant as x and y take different values, then x is said to vary
directly as y.

If - = ^, then x = ky OY y = -x and it is evident that if x
y k^

varies directly as y, then y varies directly as x.

54. The statement that x varies as y is sometimes written
xocy^ but it must be borne in mind that this is merely a
method of writing x = ky.

Example. If x — ky where k is constant, what change
takes place in y when x becomes twice as large ?

k

55. Definition. If the product xy = k ov x = - where k is

y

constant, then x is said to vary inversely as y and is sometimes
written xcc—.

y

If X varies inversely as ?/, then it is obvious that y varies
inversely as x.

Example. If xy = k where k is constant, what change
takes place in y when x becomes seven times as large?

Definition. \i x = kyz where k is constant, then x is said to

y2ivy jointly as y and z.

GRAPHIC ILLUSTRATIONS

56. Direct Variation. In case
of the straight line

y = kx or ^ = k,

X

the ordinate is k times the abscissa
for each point on the line, as has
been seen in Chapter I.

RATIO, PROPORTIOX AND VARIATION

Y

29

*X

57. Inverse Variation.

Figure 9 is the graph of

tlie equation

I. ^

xy = fc or x = -

and is known as an equi- ^ \
lateral hyperbola.

58. Joint Variation. The
area of a triangle, which is
equal to one half the length
of the base multiplied by the
altitude, is an illustration of
joint variation.

Denoting the area by x^ the altitude by y, and the base by
3, we have x = -yz.

\i x — -^^ a: is said to vary directly as i/, and inversely as z.

59. Theorem. Ifxccy and y^z then xccz.
For X = k-^y^

y = Jc^z,

(1)

(2)

where k^ and k^ are constants. Substituting for y in (1)
from (2) we have x = k k z

or

X QC Z,

60. Theorem. If xccy ivhen z is constant^ and xccz ivhen
y is constayit^ tJiefi xac yz ivhen y and z both vary together.

Let x\ y\ z' be simultaneous values of :r, ?/, z. Let z
remain constant as y changes to y\ then x must assume some
intermediate value X such that

X

y

(1)

30 COLLEGE ALGEBRA

Now let y' remain constant as z changes to z\ tlien x will
pass to the value x^ so that

-, = -,■ (2)

X z

From (1) and (2) we have

X _ yz

X y z

x^
or x= -— • yz,

y ^

where by hypothesis —j—^ is constant and therefore

y z

xozyz.

61. EXAMPLES

1. If xaz~, and if a: = 2 when ?/ = 3 and ^=1, find x
when ^ = 4 and z = 5.

2. If xccp-hq, pccy, ^oc— , and if when ^ = 1, a; =18,

and when y = 2, x = 19|, find x when y = 11.

3. li x-\- y ccx — y, show that x'^ -\- y^cc xy.

4. If X varies directly as u and inversely as v, when
uccx(^x -{- y^ and v Qcxy(x-{- ?/), prove that :?; varies inversely
as y.

5. If x^-\-2y'^azxy, and a:=l when ^ = 1, show that x
varies as y in two ways and find the ratio of x to y.

6. If the square of x varies as the cube of y. and x=2
when y = 3, find the equation between x and y.

7. If the pressure, volume, and temperature of a gas be
denoted by p, v, and t, and if p varies directly as (1 H- at')
and inversely as v, and if p = p^, v = Vq when ^ = 0, find the
relation between p, v, and t.

RATIO, PROPORTION AND VARIATION 31

8. Two circular gold plates, each an inch thick and hav-
ing diameters of 6 and 8 in. respectively, are melted and
formed into a single circular plate 1 in. thick. Find its
diameter, having given that the area of a circle varies as the
square of its diameter.

9. If a body falls from rest, the distance s passed over
(neglecting the resistance of the air) varies as the square of
the time t. If the body falls 16 ft. in the first second, what
is the relation between s and t ?

10. If a body falls from rest, the velocity v (neglecting
the resistance of the air) varies as the time t. If at the end
of two seconds the velocity is 64 ft. per second, what is the
relation between v and t?

11. What is the velocity of the falling body of Example
10 at the end of 7 sec. ?

12. If a body falls as in Examples 9 and 10, prove that
the velocity v varies as the square root of the space s passed
over, and find the relation between v and s.

13. When the body of Example 9 has fallen 625 ft., what
is its velocity?

14. Prove that when a body moves in a straight line with
constant velocity the velocity v varies directly as the space
s passed over and inversely as the time t.

CHAPTER IV
THEORY OF QUADRATICS

62. If we denote the roots of the quadratic equation

by rj, ^2, then _ j 4. V^2 _ 4

ac

^1 =

"Aa

, —h — 'Vb^ — 4 ac

and ^2 =

2a

Adding the two roots together, we have

_h-\-^'h^-4ae , -b--\/b^-4:ac
a
2b b

ri + r, = TT. + 2a

2a a

Multiplying them, we have

_ J + V^2 _ 4 ^^ _ 5 _ V^2 _ 4 ^^

1'2

2a 2a

^ l(^^h) + Vb^-4ae\\(-b)-Vb^-4:ac\
~ 4:a^

^(-5)2_(V52_4^g)2

_ 4 <3^6?_ c
4^2 ^

32

THEORY OF QUADRATICS 33

63. Dividing the quadratic by a, it takes the form

a a

or using aS^^ for r^ + rg and S^ for r-^r^^ this equation takes the
form x^ — S^x -\- iS^ = 0, from which it is seen that the sum
of the roots is the coefficient of x with the sign changed,
and that the product of the roots is the independent term in
this form of the equation.

64. Give by inspection the sum and the product of the
roots of each of tlie following equations :

1. x^-Sx-{-6 = 0. 3. 2a:2 + 4a:-3 = 0.

2. x^-h1x-3 = 0. 4. 5x^-7 X + 2 = 0.

65. To form the quadratic equation whose roots are given.
From 63 it is evident that the quadratic equation whose

roots are r^ and r^ is

x'^ - (7\ -\-r^}x-\- 7\r^ = 0,

which, by factoring the left member, takes the form

(a? — r{) (^x— r^) = 0.

If then we are given the roots of a quadratic to form the
equation, we can either take the negative of the sum of the
roots for the coefficient of x and their product for the inde-
pendent term, or Ave can multiply together the factors, x
minus one root, and x minus the other root, and place the
product equal to zero.

66. Find in both ways the equations whose roots are :

1. 1, 2. 3. if 5. 2+ V3, 2- V3.

2. - 3, 1 . 4. 3, - 1

D

34 COLLEGE ALGEBRA

FACTORING QUADRATIC EXPRESSIONS
67. Since every quadratic equation

ax^ -\-hx-\- c = ^

can be solved and written in the form

a(x— r^) Qx — r^) = 0,

it is evident that the factors of the quadratic expression
in the left-hand member of the equation are aQx— r^(x—r^.
This may also be seen from writing the expression ax^
-\-hx-\- c in the form

a[ x^-{--x-\- —- - — ^ + -
a 4 a-^ 4 a- a

^ V62-4ac\/ , h VJ2_4^^
or a[x -\- 1 ][x-\- -;

2a 2a J\ 2a 2a

_h- V52 _ 4 ^ A / _ J + V^/2 _ 4 ^^

or a\x x

2 a J \ 2 a

or a(x — r^(x — r^,

68. It is to be observed that not only expressions of the
form ax^ -\-hx-\- a but also expressions of the form ap^ -|- ap
+ 6', where p is any algebraic expression, can be factored.
Thus

(2 x^-^xy-\-4: iff - 9 (2 ^2 - 5 2:?/ -f 4^2) ^2 ^ 44 ^4
= (2 a;2 - 5 :^^^ + 4 ^2 _ 2 ^2)(2 a;2 - 5 a;j/ + 4 ^2 _ 7 ^2)

= (2 a;2 — 5 rr^ + 2 y2)(2 x^—^xy — ^ y'^)
= (2x-y^{x-2y)(2x^- y)(^x - 3 ?/).

THEORY OF QUADRATICS 35

69. Since every quadratic equation in which the coefficient
of the second power of the unknown is unity can be written
in the form ^^^ _ ^ ^ (^ _ ^^^ ^ q,

we may make use of this fact to solve quadratic equations
whose left-hand members can readily be factored. For we
have but to apply the obvious theorem that if the product of
two or more factors is equal to zero, at least one of the factors
must be zero, and its converse, that if any factor of a product
is zero the product is zero, and hence any value of x which
makes any factor zero gives a solution of the equation.

Thus a;2-2:-12 = 0,

or (x-4:){x-\-o) = 0,

and therefore the roots are 4 and — 3.

EXAMPLES

70. Factor the following :

I, 2x^-x-S. 2. 2:2 - 24 a: - 640.

3. 2(a:H-2)2-T(.r+2) + 3.

4. (a^^Qx-\-Sy-\-S(x^ + 6x + S}-\-12,

5. (x^ + x-{-iy-(ix'^-^x + l')-2.

Solve by inspection the following :

6. x'^-{-2x-S5 = 0. 7. x^-7x-S0 = 0.

8. 6 2;2-7 2;+2 = 0.

Solve by factoring :

9. (:r2-22:+ 3)2-13(2:2- 22^+3) + 22 = 0.

10. If the expression 2-2 — 3 2: + 1 has the two values 2 and

— 3, find the equation in x.

11. If the expression 0^ -\- 2 xy — 2 y^ is equal to ?/2 and

— 3 ?/2, find the equation in x.

36 COLLEGE ALGEBRA

NATURE OF THE ROOTS OF A QUADRATIC

71. It will be seen from the roots of the quadratic
equation

_J4.V62-

- 4ac

2a

_ h - V^>2 _

- -iac

5

2a
that they are

(1) real and distinct if 5^ > 4 ac^

(2) real and equal if 5^ _ 4 ^^^

(3) imaginary if 5^ < 4 ac.

In other words, the conditions that the roots are real and

distinct, real and equal, or imaginary, are that }p'=^\ao
respectively.

The roots will be rational or irrational according as 5^ — 4 ac
is or is not a perfect square.

72. If (? = 0, the two roots become

or 0,

2a

-h-VP h

and or

2a a

In other words the condition that one root should be zero
is that c should be zero.

If (? = 0, the equation takes the form

x(^ax + 5) = 0,

showing that a; is a factor of the left-hand member.

The condition that two roots should be zero is that 5 = 0 as
well as c = 0, as may be seen from the roots themselves.

THEORY OF QUADRATICS 3T

Under these two conditions the equation takes the form

ax^ — 0,

showing that oc^ is a factor of the left-hand side.

73. If we multiply both numerator and denominator of
the root

2a

by — 5 — V52 — 4 ac^ we have

_ ^ + V^>2 _ 4 ^^ _ ^ _ V52 _ 4 ^^ ^ ( _ 5)2 _ (^,2 _ 4 ^g^)

2 6'

5 _ V52 - 4

ac

Similarly multiplying the numerator and the denominator
of the root

— h — V52 — ^ac
' 2^

by —h-\- -\^b'^ — 4 ac^ we have

2c

_ 5 4. V62 - 4 a(?

If now a = 0, the roots become

2c c .2c

, or — -, and — -, or 00 .

-2V b 0

If both a and h are equal to zero, then both roots are
infinite.

In other words, the condition that one root is infinite is that
a = 0, and the conditions that two roots are infinite are that
a = 0 and that 5 = 0.

38 COLLEGE ALGEBRA

74. If 5 = 0, while a and c are not zero, the two roots
become equal in numerical value but opposite in sign. If
a= c^ the roots are the reciprocals of each other, for then

r.r^ = - = 1.
a

These conditions for infinite roots might have been ob-
tained by writing x = - and applying the conditions for zero

roots for the quadratic in ?/.

75. The expression 5^ — 4 ac is called the discriminant oi
the quadratic.

Statements analogous to those which have been made con-
cerning the nature of the roots of a quadratic equation can
be made concerning the nature of the factors of a quadratic
expression.

76. The roots of an equation /(a^) = 0 represent geometri-
cally the points on the 2;- axis where the curve ?/ =f(x)
meets it.

For example, — 1 and 3, the roots of the equation

2;2-2a;-3 = 0,

represent the points on the :r-axis where the curve
y = (x-\- 'V)(x — 3) meets it.

It is seen that to obtain the coordinates of the points is
the same as to solve the two equations

y = x^ — 2 X — ?>^

and y = ^'

77. The geometrical meaning of the foregoing conditions
as to the nature of the roots of a quadratic may be illus-
trated by the following example :

x^— S X i-p.

THEORY OF QUADRATICS

39

Here h^ — 4:ac = 9 — 4p and according as 9 — 4 p = 0, i.e.

<

according asjt?^^ the roots are real and distinct, real and

equal, or imaginary. I

As we have just seen, the roots of the equation represent
the points on the axis of x where the curve

meets the line

i/ = x^— Sx-{-p
y = 0.

Since any increase in the value of p lengthens all the
ordinates by the amount of that increase, it elevates the
whole curve with respect to the 2;-axis.

If now JO = 2, the curve cuts the axis of x at the points
x=l and a; = 2. This is the case of real and distinct roots,
Fig. 10.

>x

Fig. 10.

If p is increased to |, the curve is elevated and the two
points in which it meets the axis of x come to coincide at
a:=|, and the 2:-axis is tangent to the curve. This is the
case of equal roots. Fig. 11.

40

COLLEGE ALGEBRA

>X

Fig. 11.

If p is given a value greater than |, the curve is so
elevated that it no longer cuts the axis of x. This is the
case of imaginary roots, Fig. 12.

>x

Fig. 12.

THEORY OF QUADRATICS 41

78. Theorem. Every quadratic equation^

ax^ -{-hx-{- c=0^ where « ^ 0,

has two and only tivo roots.

It has already been shown in 62 that it has two roots, viz.

'^i=-^A

h . V52_4

ae

2 a 2 a

_ h V52 — 4: ac
^2~ ~2~a 2"a

If possible suppose that it has another root rg, different
from r^ and r^' Then we should have

ar^^ -{-br^ + c = 0, (1)

ar^^ + 5r2 + ^ = 0, (2)

argS + hr^ + c=0. (3)

Subtracting (1) from (2) gives

or a(r2 + r^) H- 6 = 0. (4)

Similarly subtracting (1) from (3) gives

«(^3 + ^i) + 6 = 0. (5)

Subtracting (0) from (4) gives

which is impossible, since a=^0 and r^ — r^^O by hypothesis.
The supposition, therefore, that the equation has a third
root different from r^ and r^ is false, and the equation has
two and only two roots.

42 COLLEGE ALGEBRA

Another and shorter proof of this proposition is the
following :

If r^ and r^ are two roots of the quadratic equation

ax^ + bx-\- c= 0,

we have seen in 67 that it may be written in the form

a(^x — r^')(x — r^) = 0.

If now rg is a root, it follows that

and therefore either r^ — r^ = 0, or r^— r^ = 0, and hence r^
is not different from r^ or r^.

79. EXAMPLES

1. Find the sum and product of the roots of the equation
(a + 2 5)a;2 - ^bcx-{- c-2 d = 0.

2. Find for what values of \ the equation

ax^ — 2 b\x + 3 (? — X = 0 has equal roots.

3. For what values of m are the roots of

mx'^ H- (a 4- 2 m)x + (3 c + m) = 0 imaginary?

4. If the equation 5 a; — 10 = 0 is regarded as a quadratic
equation, what are its roots ?

5. For what values of X is a root of

(a2 - X2)^2 -.2a\x + \'^-\-Sa^=0 infinite ?
What is the other root ?

6. What is the nature of the roots of the equation
Sx^-12x+5 = 0?

7. What is the nature of the roots of the equation
2a;2-32; + 2 = 0?

THEORY OF QUADRATICS 43

8. For what values of a and h are the roots of the equa-
tion (3a + 5-2)a;2+(2a-3^-l)a; + a + 2^ = 0 both in-
finite ?

9. For what values of a and h are both roots of the equa-
tion (a -f- 2 5) :z;2 — (3 a — 5 ^ + 5) a; 4- (« — 2 5 + 4) = 0 equal
to zero ?

10. When a is any square number, prove that the roots
of (a — 1) a;^ — 2 (a — 2) a; H- « — 4 = 0 are rational.

11. What is the nature of the factors of 2^:^— 72; + 2?

12. What is the nature of the factors of

(2^2 _ a: + l)2_4(a;2 - a; + 1) + 3?

13. The sura of two numbers is 30 and their product is
221. What are the numbers ?

14. The expression 2 a:^ — 3 a; — 3 has two values, the sum
of these values is — 5 and their product is 4. Find x.

15. The sum of the two values of a^'^ — 3 a; + 1 is — 1 and
their product is — 12. What is the nature of the values of a; ?

16. Determine x so that the sum of the two values of the
expression x^ -\- x—1 may be 5 and their product 6.

CHAPTER V

FACTOR, IDENTITY, AND REMAINDER THEOREMS

80. Factor Theorem. If P represents the polynomial of the
nth degree ^ „_i n-2 _i_ . «-3 , .

which vanishes when a; = « , then x— a is a factor of P,

Since a^a"" + a^«"-i + a^a'^''^ + a^a'^~^ + . . . + ^^^ = 0

by hypothesis, we may subtract this expression from P with-
out altering its value.
We have

P = «^2:" -t- a-^x^^''^ + «2^""^ + • • • + a„_ jO; + a,^

— (^0^^" + a^aJ"-^ + a^a''-'^ + \- a,,_^ci + a^)

= ao (x^ - a'') + ^1 (x""-^ - «"-!) + «2(^""^ - «''"^) + • • •

and as every term of this expression is divisible by x — «, it
follows that P is divisible by a: — « according to 353. This
theorem is known as the factor theorem.

Example. 2x^-\-x — 1 vanishes when 2:= — 1, for it
then becomes 2 — 1 — 1 = 0. Hence x-\-l must be a factor.
Similarly, the same expression vanishes for x = ^. Hence
a; — J is also a factor. In fact

2 a;2 + a; - 1 = 2 (a; - 1)0-^ -}- 1).
44

FACTOR, IDENTITY, AND REMAINDER THEOREMS 45

81. Number of Roots of an Equation. The proof of the
fact that every equation has a root is not simple and may
be found in Chapter XIX.

Assuming this fact or, what is the same thing, that every
polynomial vanishes for at least one value of the unknown,
it is easy to show that every equation of the nth degree has n
roots.

If «j is a root of the equation

fn(x) =a^pf^ + a-^x"~'^ + a^x"'-'^ -f ... -f a,, = 0, where ^q^^O,

then by the factor theorem x — «j is a factor of the left-hand
member, so that it vnd^y be written

aJx-+^x--^+---^^A=a,(x-a^Y,_^{x\
\ a^ a^j

where /„_-^ (a;) is a function of the {n — l)th degree.
By the same id^oX f n_^{x) has a factor x— a^, so that

fn (^) = S (^ - «l)(^' - «2)/«-2(^) •

Again

fn W = «o (^ - '^l) (^ - «2) C^ - «3)/« -3 C^)'

and finally

fn (^) = ^o(^ - "l) (^ - «2) (^ - '"'3) '-'{X- «.) •

Therefore /„ (2:) vanishes for the n values, «i, «2' H'> "*' "»•
This polynomial cannot vanish for more than n values, for

if possible let it vanish for x = P where yS is different from

each a.

Then a^ (^ - a^) (^ - a^) • • • (^ - «„) = 0,

which cannot be true unless some one of the factors yS — (x-k = ^^
or /8 = a^., which, by hypothesis, is not the case. The theorem
of 78 is a special case of this.

46 COLLEGE ALGEBRA

82. Identity Theorem. If a polynomial of the nth degree
vanishes for more than n different values of x^ the coeff dents of
every power of x vanish.

For if a^ ^ 0, the polynomial cannot vanish for more than
n different values by the preceding theorem, but by hypothe-
sis it does vanish for more than n values and therefore

"0=0-

Our polynomial is now of the (?^ — l)th degree and
vanishes for more than n— 1 values of x and therefore a-^ = 0.

Similarly ^2 = ^3 = ^4 = • • • = ^« = 0? and the polynomial

0x''-i-0x''-^ + 0x''-^-{ hOa^ + 0.

This is seen to be identically zero or to vanish for all values
of X.

Thus it is seen that if a polynomial of the nth degree in
X vanishes for more than n values of x^ there are three
equivalent ways of stating the conclusion: 1. The coeffi-
cients of the various powers of x vanish. 2. The polynomial
vanishes for all values of x. 3. The j)olynomial vanishes
identically. This theorem is called the identity theorem.

83. Identity of Two Polynomials. If two polynomials of
the nth degree are equal to each other for more than n values
of x^ the coefficients of the corresponding powers in the two
polynomials are equal.

Let the two polynomials be

a^x^ + a^x'^~'^ -\- a^x^^"^ -\- -•' + cin
and h^x^ -^ h-^x''-'^ + h^x''-'^ -\ \- h^.

Then if these are equal for more than n values, it follows
x^ (^0 - 5o) + 2;«-i (aj - 5i) + • • • 4- {a, - h,)

FACTOR, IDENTITY, AND REMAINDER THEOREMS 47
vanishes for more than n values, and therefore

a

(j-5o=0, ai-^i = 0, .-., a„-hn = 0, (82)

or «o = ^0, «i = ^1, • • •, cin = h^.

Let the student state two other forms for the conclusion
of this theorem.

Example. Find the condition that a^x^ + 2 a^x + ag may-
be a perfect square.

If the expression is to be a perfect square, the square of
ax+ yS say, we must have

a^py^ + 2 a^x -\- a<^= {iix + ^y-

therefore o^ = a^, «/3 = a^, ^ = a^j,

whence d?'^ = a^a^, also (^a/sy = a^/3- = a^,

and therefore a^a^ = af^ or -5 = -^ •

The student will see that this agrees with the conditions
of 71 that 4 a^2 _ 4 ^^^^ _ q, or a-^ = a^^g-

84. The foregoing gives us a convenient means of finding
by trial the factors of some polynomials.

Thus, in p = ^3 _ 6 ^2_,_ 11 ^_ 6,

if we put x = l we have

l_6 + ll-6=0,

and therefore rr — 1 is a factor.
If we put x=2 we have

8 - 24 + 22 - 6 == 0,

and therefore a; — 2 is a factor.

48 COLLEGE ALGEBRA

Again, if we put x= S, we have

27 - 54 + 33 - 6 = 0,

and therefore a^ — 3 is a factor.

Since P is of the third degree in x there can be no other
factor involving x, and if there is any other factor, it must
be numericah Denoting it by iV, we have

a^-6x^-hllx-6 = N(x-l}(x-2)(x-S),

and since this is an identity the coefficients of like powers

of X on both sides must be equal. Equating the coefficients

of a^ we have

l=iV.

Therefore P = (x -l)(ix - 2)(x - 3).

85. Remainder Theorem. If a polynomial

P =/„ (x) = a^x^ + a-^x"^'^ + a^x""'^ +•••+««

he divided hy x — a, the remainder will be the result obtained by
substitutifig a for x in P.

Let Q be the quotient which Ave obtain on dividing P by

X— a until we get a remainder P, which does not contain x,

so that ^ . .

P=Mx} = Q(x-a)-^E,

which is an identity, true for all values of x, since it is true
for more than n values of x different from a, and hence true
for x = a.

Placing x= a, we have

/«(«) = ^o^"" + ^i""""^ + ^2^""^ +.-. + «,, = 72.
Tliis is called the remainder theorem.

FACTOR, IDENTITY, AND REMAINDER THEOREMS 49

86. EXAMPLES

Factor :

1. :c3 + 5a;2- 9a:-45. 3. r^ -\-2 x^ - 2Sx- 60,

2. x^-a^-S9x^-^2^x-\-lS0. 4. x^- ^x"^ - x+ 6.

Find the remainder when :

5. a^ — Sx^—5x-\-lis divided by a: — 2.

6. xi^ — 2x'^-\-Sx—o is divided by 2^ + 4.

7. Ux^-^x^+2x-l = ax'^-hC2a + b)a^-\-(^b-c)x'^

-i-(c-2d)x-d-\-2e,
find the values of a, 5, c, d, e,

8. Prove without assuming the result, that -O = -i = -2 is

the condition that a^oi^ H- 3 a-^x^ + 3 a^x 4- a^ may be a perfect
cube.

9. Find the condition that 2:^ + 3 Hx + (r may contain a
square factor (x + «)2 .

Suggestion. Assume x^ -\-?> Hx-\- G = {x-\- ay-^x + (i).

10. Show that the values of the 5's must all be unity in
order that

Factor :

11. x^-Qs^ + ^x^+12x-m.

12. x^-6x^-\-\lx-Q.

13. :?:*+22:3_3^_4^4.4,

14. 2;4-10.T3 + 35a^- 502-4-24.

15. a%-\- ah'^ -\- IP-c + ?>c2 + c^a + a^c-i-2 ahe.

16. a^h-\-ab''-'-{-h'^c + hc^^c'^a-\-a^c-{-^ahe.

CHAPTER VI

COMMENSURABLE, INCOMMENSURABLE, AND IMAGINARY

NUMBERS

87. Definitions. The work in elementary algebra involves
positive and negative integers, zero, and positive and negative
fractions. These numbers are all known as commensurable
numbers.

Suppose we are given x^ = w, where m is any commensura-
ble number. Then x is called the square root of w, and may
be defined as that number which when multiplied by itself
will produce m. It is represented symbolically by x = Vm!
Three cases will arise :

First, suppose that m is a perfect square of some commen-
surable number, so that m = t^. Then our definition of x
will permit it to have either of the values, ^ or — ^. Thus
it will be seen that the square root of m has two values.
For the sake of simplicity, unless there is some particular
reason to the contrary, the symbol Vm is taken to represent
the positive root, so that the two values of x are represented
by Vm and — Vm, or by the combined symbol ± Vm.

Second, suppose that m is positive, but not a perfect square,
as, for instance, 2. Then no commensurable number can
possibly represent V2, though by taking a sufficient number
of decimal places, a number can be found which will satisfy
the conditions within any prescribed degree of accuracy of
approximation.

50

IMAGINARY NUMBERS 51

For, if possible, suppose that V2, which is equal to
1.4142135- ••, as may be found by trial, is a number whose
decimal part terminates. Then, squaring both sides, we
have an impossibility, as the right-hand member cannot
be 2. Hence the supposition that the expression V2 is a
terminating decimal is false. Neither can V2 be expressed
by a commensurable fraction. For, if possible, suppose that

V2=-, where - is a commensurable fraction reduced to its

0 0

lowest terms. Then, squaring both sides, we again have an
impossibility, as — is a commensurable fraction reduced to

its lowest terms and cannot be equal to the integer 2.

Such a number, whose value cannot be exactly expressed,
either by an integer or by a fraction, but which can be ap-
proximated within any prescribed degree of accuracy by using
a sufficient number of decimal places, is called an incommen-
surable number.

Third, suppose that m is a negative number, as, for instance,
— 3, or — 4. The principles of elementary algebra show that
no positive or negative number when multiplied by itself can
possibly produce a negative number. Hence if we are to
represent a true solution of the problem, we must introduce
a new kind of number. Thus we define V— n^ Avhere n is
intrinsically positive, as the number such tliat (±V — n)^
= — n, and such a number is called a pure imaginary. It
obeys all the laws of algebra, and its properties will be ex-
plained in the last division of the present chapter.

In contrast with the pure imaginaries, those numbers which
have been defined as commensurable and incommensurable
are called real numbers.

A combination of real and imaginary numbers in the form
a±V— 6, where a is real and b is intrinsically positive, is

52 COLLEGE ALGEBRA

called a complex iiiimber. The properties of complex num-
bers will be discussed in connection with those of pure
imaginary numbers.

THEORY OF INDICES

88. The proof of the fundamental theorems of the theory
of indices for positive integral exponents * are given in
elementary algebra, but for the sake of completeness the
following are reproduced here:

If m and n are positive integers, and x is any commensu-
rable number, ,^ .,.,.. -i- -c ^

' x^^ = X X X X X X ••• to m tactors,

and x^ = xxxxxX'--to n factors.

Therefore x"^ xx'^ = (x x x x x x • • • torn factors)

x(xxxxxx---ton factors)
=^ X XX X X X ■•• to (m + 7i) factors

Applying the above principle twice, we have

^7«i X 2:'"2 X a;"'3 = (af^^ X x"^^~) X x^^

Continuing the process, we have

If, now, m-^ = m^ = m^= -" = 771^ = ^1,

this becomes x"" x x"^ x x'" x ••• to n factors,
or, (x"y' = x""^

Corollary. Likewise (^x'^y = a^'"" = x"""".
Therefore (x'^y = (x'^y.

* In this book the terms index and exponent are used interchangeably.

IMAGINARY NUMBERS 53

If X and y are two commensurable numbers, and ri is a
positive integer,

rjiaiyn =z(xv.xy.xy.--'ion factors)

x(t/ X y X y X •■' to n factors)
= (xy) X (xy) X (xy) x ••• to n factors

Finally, if m and n are positive integers, and m > n^

„ rr'" x X X X X X •••torn factors
x'^ X XX XX X "• to n factors

= X X X XX X ••• to (m — n~) factors

^m—n

89. In 88 it is assumed that the exponent n is a positive
integer. In the remainder of this chapter, the meaning of
x"^ will be extended so as to allow n to have any commensu-
rable value. Consistency requires that the extension shall
be made in such a way that the laws for positive integral
exponents, as heretofore developed, and the other funda-
mental laws of algebra shall be obeyed. It will be shown
that if all the new exponents are defined by the requirement
that they must satisfy the law x"^x'^ = a:"'^", for positive
integral exponents, then they will satisfy all the other laws

for positive integral exponents.
1

90. Since a;% where n is a positive integer, must obey the

same law as 2:% where 71 is an integer, we must have

111 1

x'^ • a;" • a;" ••• to ^ factors, or (2:")", equal to

-+-+-+••• to n terms

From (a;") = x^ we have x^ = Va;, by taking the nth root of
each side.

54 COLLEGE ALGEBRA

11 1

Suppose x{^x^ ' ■ ■ x,^ = ci",

1

and (^1^2 • • * ^'n)" ~ ^'

Raising each side of both equations to the Tith power, we

get x,X.

Therefore a" = 5% or a = 6,

111 1

Raising both sides of this to the pih. power, we have

p p p p

♦*'l •*'2 " ' »t — V'*'l'*^2 *^/ny •

Therefore positive fractional indices obey the law

the same as a positive integer n.

91 . In 90 suppose that each of the m factors x^x^ ••• Xj^ is
equal to x, then, i l ^

(cc")"" = (x'^y = x^,
1 _

or, since x"^ = "v/a:,

m

therefore ( V^;)'" = V:r'" = a:"^,

which shows that the m\h power of the n\X\ root of a number
is equal to the n\h root of the mth power of that number,
and that both are equal to that number with a fractional

ryyh

exponent — •

n 11

Again, let {pc^y = a,

1 1
and {xT'y' = 6.

IMAGINARY NUMBERS

5b

Then,

and

Therefore

1

x"" = 6% and x = b'"

^mn ^ f^mn^ OT a = b,

Again,

Therefore

which shows that the nth root of the mth root of a number
is the same as the mth. root of the nth. root of the same
number.

x'i = (x'ly.

(^p and q being positive integers)

p m 11

(m and n being positive integers)
1 1

pm

1 1

1 1

= [\(x"y}'^y

m p

= (x^y.

It is therefore seen that positive fractional indices obey

tne law fry,m\n ^ r^'n\m _. ^mn

the same as positive integers m and n.

Since the zero exponent must obey the law, x"^x^ = a:"^"*"", the
same as positive exponents m and n,
we have '**"* ■ '^^ — ryju+o

X"

X^ = X"

X

= X"

:rO = — = 1.

X'"

Therefore

Likewise negative exponents must obey this same law.

56 COLLEGE ALGEBRA

Therefore a;™ • x~"^ = a;'"-'" = x^ = 1 .

Hence x~"^ = — .

Therefore any number (zero sometimes excepted) raised to
the zero power is equal to unity, and any number with a
negative exponent is equal to the reciprocal of that number
with exponent positive.

The negative exponent also obeys the law (2:'")^= (a;")'" = a:^"".

For (a:-"0" = ( — Y = — = ^"""' = -^— = (x"") -™.

Also (x'^y = ( — ) " = -i- = x"'"" = — - — = (2:-")-^.

Likewise the negative exponent obeys the law
For

Hence we have shown that by requiring all new exponents
to satisfy the law x'^'x^ = x"^'^^, it follows as a consequence that
they also satisfy the laws

and rri":r/ ••• xj = (^x^x^ ••• x„,y.

92. EXAMPLES

Express with radical signs :

2 \ 3 5 2. ? r

1. a^b". 2. rri^n^p^. 3. xiy^

Express with fractional exponents:

4. ■\/a^W. 5. V^\/P^^. 6. V^V/.

IMAGINARY NUMBERS 57

Express with positive exponents :
7. 7?y~^z~^. 8. a~%\c~^, 9. vrhi'^.

Express without denominators :

10 ^^^^ 11 m^^n^ 8a

a^'¥c'

Find the values of the following :

13. (a^)6. 15. (m"%^)i^ 17. (8^)5.

14. (0:^^)^. 16. (24^)1 18. (-729)1

Multiply the following :

19. 5 a;^ — a:^ + 7 by 2 a:^ + 3 ^3 _ 4,

20. 2 r?:* _|_ 5 ^3 _ 3 ^-2 + 7 by 3 ^-3 _^ 7 ^-2 _ n,

21. 3 c^h^ - 7 a^5^ + a-W by 2 a^r^ - 3 a%~^.

Divide the following :

22. 2:^ + 372:^-700:^ + 50 by a;^- 2 r?;^ + 10.

23. 8a-^^-8a^ + 5«=''^-3«-3'^ by 5««- 3a-^

24. 5 6^ - 6 5 - 4 - 4 5* - 5 jHy 5^ - 2 6i

RADICALS

93. Definitions. Any number to which a radical sign is
attached is called a radical ; as V2, V5, V9.

If m is a commensurable number which is not a perfect
wth power, -\/m is incommensurable, 87. It is then called
a %urd. A %urd is thus defined as an incommensurable root
of a commensurable number. Thus V2 = 1.4142- •• and can-
not be exactly expressed, however far the calculation is
carried out.

58 COLLEGE ALGEBRA

It will be observed that all surds are radicals, but not all
radicals are surds. Thus V9 is a radical, but not a surd.
V^ is or is not a surd according as a is or is not a perfect
square, but in algebra all such expressions as Va, VS, etc.,
are considered as surds.

Any number which can be expressed without involving
surds is said to be rational.

All surds or numbers which cannot be expressed without
involving surds are said to be irrational.

The order of a surd is the index of the root which it repre-
sents. Thus a surd which is a square root is of the second
order, and one which is an nth. root is of the nth order.

It has been shown that surds can be written as fractional
powers of numbers.

A surd or radical which has a rational coefficient is called
a mixed surd or mixed radical.

A surd or radical which has no rational coefficient except
unity is called an entire surd or entire radical.

94. Three types of surds and radicals demand special
attention :

1. Those in which the expression affected by the radical
has a factor which is a power of the same degree as the index
of the radical ; as V250, -\/a^h.

2. Those in which the expression affected by the radical
is a power whose index is a factor of the index of the radical;
as V8.

3. Those in which the expression affected by the radical
is a fraction ; as V|.

95. Since it has been shown in 90 that surds and radicals
can be expressed without the radical sign by means of frac-
tional exponents, the laws of exponents furnish two princi-
ples which are of use in the reduction of surds from one form
to another.

IMAGINARY NUMBERS 59

1. ^~ab = Qahy = a^f^=-^a-</h,

m 1

2. "'■VaJ^' = a^' = a^' = </a,

These principles permit a transformation of the three types
of 94. Thus,

■^250 = ^125^2 = VJ2b . V2 = 5^.

-^8 = ^23 = V2.

Since the denominator in the last case must be a perfect
square, the first step must be to multiply both numerator
and denominator by 3.

These reductions show that it is permissible to define a
surd as in its simplest form when the expression affected by
the radical is integral and as small as possible.

96. EXAMPLES

Reduce the following to their simplest form:

1. Vl2. 5. ^/32. 9. -Vimba^y^.

2. Vl8. 6. V4a. 10. V«2 - 2 a6 + ^2.

3 /TT^

3. V32. 7. V27 a%. 11. V2 x^ + 4:xy-{-'l y^.

4. V24. 8. V50a362. 12. V27 2;2 - 81 i/2.

13. 4a;V(38^y. 16. V64a4.

14. (9 a2_ 27^2)1^ . 17^ 8^25:^14^^-6.

15. (16a3-48a25 + 48«52_i(3 53)l. 18. J^ a/64000 o^y.
19. Vi. 20. V|. 21. V|.

60 COLLEGE ALGEBRA

The student will here notice that it is unnecessary to mul-
tiply both numerator and denominator by the entire denomi-
nator, but only by such a factor as will make the denominator
a perfect square.

Thus V 7 — VI5 — V-J- • 14 = V^ • Vl4 = 1 Vl4

44. ^ V2-0 0-

Here the denominator must be a perfect cube, and the
smallest permissible multiplier is therefore 5.

TTpnpp 2 V -2-1- — 9 -^/JLOA^ — 9 V - J V 105

= 2- J^V105 = -1V105.

23. \—T7y — • 27. \ 7^^-

^ 3 ^c^ 0? — y^^ y^

— X

x-\- y l + a;^(l + xy

97. To reduce a mixed surd to an entire surd. This is
another application of the same principle as 95.

Thus, 2 ^ = \/23 . ^5 = -v/iO.

Also, I V7 = V| . V7 = VM.

Reduce the following to entire surds :

1. 5V10. 3. — 7 \— n:-

a — 0 ^ a-\- 0

2«3/-2T2 ^ ^-^^ ^

2. ^^<Ja%\ 4

J^^-f^

3 5 a;4-y ^'

IMAGINARY NUMBERS 61

98. The problem of reducing a number of surds to the
same order may be illustrated by writing them with frac-
tional exponents, and then reducing these exponents to a
common denominator. Thus, if it be required to reduce V2,
V 5, V7, and VTO to the same order, we have

^/2 = 2^ = 2'^ = '-</¥ = ^■M;
^5 = 53 = 5^ = ^5^4 = ^625;
^7=7^ = 7r-=^r3 = ^MB;
-^10 = 10* = 10^^ = ^102 = ^100.

EXAMPLES

Reduce the surds in each of the following groups to the
same order :

1. </5 and -^7. 5. ^(a - by and </{a + by.

2. VT and -v/TO. 6. </2, ^10, and ^5.

3. V^ and </(^d. 7. V|, ^'f, and </^.

4. V2Wc and Vchl^. 8. ^|, -\/a%% and V2 a%.

Note. In Example 8 the student will see that though Va-b^ is
nominally of the fourth order, it is equal to Vcib, and hence should be
regarded as of the second order. Hence the three may be reduced to the
sixth order, instead of the twelfth, as might perhaps be supposed.

9. ^'l', \§, ^i o\ and ^5 d3.

10. \/— ■) \'-, \x^y^ and v

y 'X

Definition. Similar surds are surds which, when reduced
to their simplest form, do not differ at all, or differ only in
their coefficients.

62 COLLEGE ALGEBRA

ADDITION AND SUBTRACTION OF SURDS

99. For reasons which will become apparent in the con-
sideration of the properties of quadratic surds, only similar
surds can be united into one surd by addition or subtraction.
Therefore, in order to add or subtract surds, reduce each to
its simplest form, and combine the coefficients of those which
are similar.

100. EXAMPLES

1. Add V6, V2i, Vf, V15, V|.

Reducing to the simplest form and uniting similar surds,
we have

V6+VM + V| + Vl5 + V|

= V6 + 2V(5 + lV6+Vl5 + -iVT5

= (1 + 2 + 1) V6 + (1 + -J) Vl5 = JgQ- V6 + f VT5.

2. Simplify

VWM 4- V50"^ _}_ 2 aVJh -7</¥- 3 Vl8^.

</WM 4- V50 a%^ 4- 2 a</Sl) -1</¥ - 3 Vl8^2
= SaVb-h5ab VWa + 4. a</b -1 h</b - 9b V2^
= (3« + 4«-75)\/6 + (5a5-9 J) V2^
= (7 a - 7 ^)-v/6 + (5 a5 - 9 J) V2^.

Simplify :

3. V8 + 2VT8-3V32-14V2.

4. V20 + V45-7V|.

5. 3V2a + 6V54a;-7aV16.

6. 4v'32-5-v/162 + 14a/8T.

IMAGINARY NUMBERS 63

11. , V7 2;« + 56 x* + 112 a^ - V3 13 - 18 x2 + 27 2;.

12. •</1876^V^ + 2a-!/v'48l?p-6^y-^|^

13. ^2r^ + ^24-A^|' + ^i.

14. 4 a; V68 x^y"- + 5 «/ VlT :r/ + 14 V60 xif.

«/12 y

16. V2 2:2 + 4 2-?/ + 2 ?/ + VI8 ?/2 _ 12 :?;j/ + 2 x^

-V2a;2-8a;?/ + 8/.

MULTIPLICATION OF SURDS

101. Two methods are in common use for the multiplica-
tion of one surd by another. They are exemplified in the
following :

1. V2 X V3 = V2^<~3 = V6, (95)

or V2x V3 = 2^x3^=6^ = V6. (90)

In this example both methods are exceedingly simple, being
nothing but direct applications of the elementary principles.

2. V2x^2=^8xa/4=^32;

or V2 X a/2= 2* X 2^ = 2'-^* = 2'^ = -^ = ^32.

For the first method in this problem it is necessary to re-
duce the surds to the same order before 95 can be applied.

64 COLLEGE ALGEBRA

The second method is a direct application of 89, since the
quantity affected by the exponent is the same in both
factors.
3.
■^6 X -v/12= ^1296 X ^1728 = a/1296 x 1728 = ^2289488,

or a/6 X v'12= 6* X 12^ = 6^'^ x 12^'^= (6*)^'^ x (U^y'^

= (1296y'^ X (1728)T^
= (2239488)!'^

= V2239488.

By the first method this example is just like 2 ; but the sec-
ond method there used will not apply in exactly the same
way, since the quantities affected by the exponents are not
the same in the two factors. Hence it is necessary so to
transform the factors that the exponents will be the same, in
order that 89 may apply.

4. 2V5x3V2 = 2V25x3V8 = 6V200,

or

2(5)^ X 3(2)^ = 6(5)^ x(2)* = 6(25)*(8)* = 6(200)* = 6-^200.

Since the product of a series of quantities is the same in
whatever order they are taken, the coefficients are multi-
plied together independently. The surds are then multi-
plied together by either method.

Simplify :

^ -^ 5. VlO X Vl5 X V6.

6. a/12 X V5 X </S.

7. 2VT2 X 3V15 X 4a/5.

8. 3Vix4Vfx5V|.

" V-^#

IMAGINARY NUMBERS 65

J2 \ ^2 \ ^2

11. Va^ — 52 X V (a + 5)2 X va — ^.

12. Multiply 2 VS - 5 V2 by 3 V3 + 4 V2.

2 V3 - 5 V2

3 V3 + 4 V2
18-15V6

+ 8V6-40
18- 7V6-40=-22-7V6.

13. Multiply 2 V3 - 5^2 by 3 V3 + 4 v^.

2V3-5^
3V3 + 4^2

18 - 15 Vl08

18- l</10S-20</4:.

In order to multiply 2V3 by 4v2 it is necessary to reduce
them to the same order.
Multiply :

14. 3V3-4V2by 2V3-6V2.

15. 3v'4-4^2by4^4-3^2.

16. 2 Va5 — 2Vcd by 2 Vac — 2^bd.

17. 3V^-2V^-4V«5by 2-v^-3V^.

19. 2Vh + SVc + 4:^d by V« + V5.

20. 3 V^ - 4 V^2 by v^ + VP.

66 COLLEGE ALGEBRA

DIVISION OF SURDS

102. The methods in use for division of one surd by an-
other involve the same principles as those used in multiplica-
tion, and are so closely related to these methods as scarcely
to need any explanation.

.— q/ — fi/ ■ fi/ ■ Q\a%^ elh 1 fi,— —

s/—or ^ab _ '\/ah -^aW- _ h-\a^b _ 1 r,—^

r^-n /~T 3/-nT \ at) W ao -\ ab^ b-\ ayb i gz-tt-
or Va6 ^ V a^6 = -^—^ = ^-=: X -ir;=L = t" = - Va^b.

In the first method the surds are reduced to the same order,
whence division is readily effected, and the result is reduced
to its simplest form. In the second method fractional
indices are used, which are reduced to a common denomi-
nator, then to a common index, whence 90 enables the
division to be effected. The result is restored to its surd
form and simplified. In the third method the division
is first indicated by writing the divisor as the denomi-
nator of a fraction. Then both numerator and denomi-
nator are multiplied by that quantity which makes the
denominator rational. This does not change the value of
the fraction. The multiplication is performed as in 95.
This last method is known as the method of rationalizing
the denominator.

IMAGINARY NUMBERS 67

Simplify :

2. V24-f- V6.

3. V24-VT6.

4. V50-V-V40.

5. V63-v'42.

6. V8^^12.

7. ^/24-V6.

8. 'Vab -^ "V^.

EXAMPLES

9.

■yah -V- Va6.

10.

V2 a6 - -\/2 a26.

11.

3^^,2^^5-.y5^,

12.

2Va-5^3Va4-J.

13.

3^1 -V|.

14.

V6 4-Vl5-^2V3.

15.

VB + V5 - 2^/3.

COMPARISON OF SURDS

103. The relative magnitude of a series of surds may be
readily compared if tliey are reduced to the same order ; thus
to compare Vo, VlO, and V32, we reduce them to the twelfth
order.

V0= ^15625; ^10 =v 10000; v'M = v'MTeS.
A comparison of these results shows that

-{/32 > V5 > ^yiO.

INVOLUTION AND EVOLUTION OF SURDS

104. A surd may be raised to any required power by any
of the methods given for multiplication of surds. Fre-
quently it is advisable to express the surd as a fractional
power of the expression affected by the radical, and then
raise it to the required power. Thus

Any root of a surd may be obtained by a similar process.
Thus 3, , 1 1 1 ,.,

68 COLLEGE ALGEBRA

RATIONALIZATION

105. Definitions. Rationalization is the process of making
a surd rational hy multiplying it hy some other expression.

The expression which is used to multiply a given surd in
order to make it rational is called a rationalizing factor.

A binomial surd is a binomial in which one or both of the
terms is a surd.

Two binomial quadratic surds are called conjugate surds if
they differ only in the sign of one of the terms.

Thus 4 - V5, x^ -i-yK 3 + V7, 3 - VT, V^ + V?, and
-\/y — Vsj are all binomial surds, and the last four form two
pairs of conjugate surds.

Since the product of the sum and difference of two quan-
tities is equal to the difference of their squares, it follows
that the product of two conjugate surds is always rational.

106. The method of reduction of a fraction whose denomi-
nator is a monomial surd to an equivalent fraction with a
rational denominator has been given in 102.

When the denominator is a quadratic binomial surd, the
process is as follows :

2 - V3 ^ (2 - V3)(3 - V5) ^ 6 - 3 V3 - 2 V5 + Vl5

3 + V5 (3 + V5)(3-V5) 9-5

6_3V3-2V5 + Vl5

Also,

■Va - 3Vg ^ (Va - 3V^)(V^+ 2V^) ^a- Vab -6h
Va-2V^~(Va-2V^)(Va + 2VD~ a-4:b

When the denominator is a binomial surd of the nth order,
the process depends upon two well-known theorems in factor-
ing, viz. :

IMAGINARY NUMBERS 69

W7ien n is odd., x^ ± y'^ has for factors x±y and x^~^ T x^~'^y

Whe7i n is eve7i^ x^ — y^ has for factors x ±y and x^~^ T x^~'^y
+ x''~^y^ T ••• T y""^ For a proof see 86, Ex. 10, and 353.

107. EXAMPLES

91
1. Rationalize the denominator of

3 --^4

Therefore

(3)3_ (^'4)3^ (3 _ ■^4)(32 + 3sy4 + (V4)2.
2

2r3^ + 3V4 + (:v4)^1 ^2[32 + 3V4 + (V4)2-|
"(3_^4)[32+3v/4 + (^4)2] 33 -(^4)3

2(9+ 3a/^4 + 2v^2) ^ 18 + 6^4 + 4^/2
"27-4 23

2. Reduce — z =: to an equivalent fraction with a ra-

4/ , 4/7 ->-

tional denominator.

^ +V^a(A/6)2-(^5)3].

Therefore

</a-\-</h

(</'ay - (</^)^</b + </a(</by - (</hy

(-^rt + ^6) [(-y«)3 - C</ay</b+ -\/^(a/^)2 - (a/6)3]

V«3 — Va^ V5 + Vrt V62 — VP
(vay-(A/6)4

->y^3 _ ^^ -I- -^^2 _ ^p

a — 6

70 COLLEGE ALGEBRA

3. Express — = ^-^ with a rational denominator.

Va;— V«/

The least common denominator of the indices 2 and 3 is 6,
and therefore, by the principle above stated,

1 nereiore, — r: ^^

Va; — "V ?/

(Vi)(^^)4 + (^^/] divided by (Vi- ^^)[(V^)5 +
( V;r)4-v^ + ( Vxf{</]/f + {-w'xfiVyf + Vx{</]/y + (^^)5]
_ a{x^^\fx-\-o^^ y^x^x^y^Ar xy 4- ^xy\y + ?/ V^^)

_ a(x^^x + :z^^ V ;^ + x^x^y^ -\- xy -\- y^.x^y'^ -f- y^y'^')

x^ — ?/2

4. Reduce - — ^ z ^ to a fraction with a rational

V2-V8 + V5
denominator.

Multiplying the numerator and denominator by V2 — V8

— V5, the fraction becomes

2-(V3+V5)2^-6-2Vl5^3 + Vl5
(V2-V3)2_5 -2V6 V(3

which is equal to

3 V6 4- 3 VTO ^ V6 + VTQ

6 ~ 2 *

IMAGINARY NUMBERS 71

In general a trinomial expression of the form V5 + Vy
+ V2 can be rationalized by multi23lying it by the expressions
■^x — V^ — V2, Vo; + V?/ — V2, Va; — V^ + V2 ; the product
of the first two is a; — ( V«/ + V^)^ = x — y — z — ^'^yz^ and
tlie product of the second two is 2; — ( V?/ — V^)^ = x — y — z
+ 2^yz^ and the product of these partial products is the
rational expression (x — y — zy^ — \ yz = x^ -\- y"^ -\- z^ — 2 xy
— 1yz—2zx. The four expressions here multiplied together
are called conjugate trinomials. It is to be noted in the fore-
going example that the product of the remaining trinomials
V2 + V3 — V5, V2 + V3 — V5 is 2 V6, which shows why V6
had to be used in completing the rationalization. In general
any polynomial quadratic surd may be rationalized by multi-
plying it by the product of all of its conjugates.

One of the objects of rationalizing the denominator of a
fraction containing surds in the denominator is for the pur-
poses of calculation, for rationalizing the denominator saves

both time and labor and conduces to accuracy.

2

5. Thus to calculate the value of we have,

V3+ V2
after rationalizing the denominator, to calculate the value of
2(V3- V2), which is equal to 2(1.73205.-. -1.41421...)
= 2(.31784...)= .63568... The student may compare this
with dividing 2 by 1.73205... + 1.41421... or 3.14626...
Rationalize the denominators of :

6.

V2- a/3 g V2+ V3
V3 + 2 ' V5 - VT

, 6-V3 a+V6
7. -' 9. — ' =•

12. ^-^+^-^-Vi. ,3.

10. ^''

-V6

- V5

11 ^-

_ V3+ V7

V2

H- V3h- VT

Va +

V3- V2

72 COLLEGE ALGEBRA

Calculate to four places of decimals the values of :

14. . ^- 15. 2yi-V2

V^-V2 2V8 + V2

16 ;^^ , given that V5 = 2.23607 ....

V5-2

PROPERTIES OF QUADRATIC SURDS

108. Theorem. A surd cannot equal the sum of a rational
expressioii aiid a surd.

For, if possible, let Va = 5 + V^

where 5 is a rational expression, and Va and V^ are quad-
ratic surds. Squaring both members, we have

r a—lP'—c
or Vc= — ,

A 0

that is, a surd is equal to a rational expression, which is
impossible.

109. Theorem. If a+V6 = tf+V5, where a and h are
rational numbers and V^ and 'Vd are surds, then a = c and
h=d.

For, by transposition

V^ — c— a-\- V(/,

and by the preceding theorem <? — a = 0, and therefore
c= a, and consequently b = d.

lia Theorem. If ^ja-\-Vb= Vx-\- V^, then yja - V6
= Va: — V?/, if a, 5, X, and y are rational, V6 is a surd,
and a > Vi and x>y.

IMAGINARY NUMBERS 73

For, by squaring we obtain

Therefore, by the last theorem,

a = x+y,

and V^ = 2 ^xy.

Subtracting a — V3 = x — 2^xy + y.

Extracting the square root,

V« — V5 = ± ( vi — v^),

where the double sign is determined in accordance with the
conditions of the problem.

111. To extract the square root of a binomial surd. The
method of doing this may be illustrated by the following
example :

Find the square root of 16 + 2 V55.

Like every other number, the binomial surd has two square
roots ; but since the given binomial has both terms posi-
tive, it follows that the positive root must have both terms
positive.

Hence we may let \16 + 2 V55 = Vo: + V^. (1)

Then, by 110, Vl6 - 2 V55 = V^ - V^. (2)

Multiplying (1) by (2), V256 -4-55 = x-y,

or Q = x — y. (3)

Squaring (1), 16 -f 2 V55 = x-\- y -\-^ V^,

whence, by 109, 16 = x -\- y. (4)

From (3) and (4), x = 11, and y = 5-

Therefore ^|l6^^^2VM = ± ( VlT + Vo) .

74 COLLEGE ALGEBRA

An examination of the above process shows that x and y
are the two numbers whose sum is the rational term 16, and
whose product is bb^ and that a similar statement must
be true of every like problem. Hence, when the numbers
involved are not too cumbersome, examples like the above
may be solved by inspection as follows :

Reduce the surd term to the form in which its coefficient
shall be 2. Then find two numbers such that their sum
shall be equal to the rational term, and their product equal
to the expression under the radical sign. Extract the square
root of each of the numbers so found, and connect the results
by the sign of the surd term.

112. EXAMPLES

Extract the square root of :

1. 4 + 2V8. 5. 12-VT40. 9. -V--^¥--

2. 4-2V3. 6. 11-V96. 10. 1 + |V6.

3. 5 + 2V6. 7. 57-12VT5. ii. 3-|V5.

4. 7 + 4V3. 8. 93-fV5400. 12. 2a-1^a^-hK

-'4

13. m + '- --/^^^

14. 11 a-Sb + W6a^-ab-b\

RADICAL EQUATIONS

113. Definition. An equation involving one or more irrational
roots of an unknown nuinher is called a radical equation.

In case of the more simple types of radical equations the
solution is readily effected by the ordinary algebraic processes.
Yet it is necessary always to be on the watch lest false solu-
tions be obtained.

IMAGINARY NUMBERS 75

1. Solve V^ + 7 = 8.

Squaring both members, rr + 7 = 64.

Hence x = 57.

This result satisfies the original equation, and therefore is a
true solution.

2. Solve Va7 + 7 = — 2.

Squaring both members, 2; + 7 = 4. (1)

Hence x= -?,. (2)

According to the laws of algebra (1) must be true if the
original equation is true, and (2) gives the only possible
value of X which satisfies (1). Therefore if there is any
true solution of the original equation, it must be a: = — 3.
But trial shows that this value does not satisfy the original
equation. Hence we are forced to conclude that there is no
true solution of that equation.

In fact (1) would likewise have been obtained if we had
started with the equation Va; + 7 = 2, and we have really
obtained the solution of this latter equation instead of the
one which was given.

Some writers avoid this difficulty by regarding all radical
signs as affected by the double sign, as ± ^x-\- 7. But this
merely transfers the difficulty to the later applications of
these principles.

3. Solve V2 x-{-l — ^x — b = 0.

Transposing, V2 x-\-l = V2; — 5.

Squaring, 2 x-\-'i = x — 5.

Hence x= — 12,

which satisfies the original equation, and is therefore a true
solution.

76 COLLEGE ALGEBRA

4. Solve Va; + 7 + -Vx-5- 10 = 0.

Transposing so as to have one radical alone in one member,

VxT^ = 10 - -Vx - 5.
Squaring, a; + 7 = 100 + a; - 5 - 20Vx-5.

Transposing and combining,

20 V^^^ = 88,
or 5-Vx-b = 22.

Squaring, 25(x — 5) = 484.

Reducing, x = 242^^.

This satisfies the original equation, and is a true solution.

5. Solve V^ 4- 7 + V:r — 5 + Va; — 2 — Va; + 6 = 0.

It is not practicable to separate one radical in this problem,
since that would leave three in the other member. Hence it
is better to leave two in each member.

Thus, Vx+ 7 + V:r-5 = V^+IT- Vx-2. (1)

Squaring, x+7 -\- x— 5 + 2Vx^-{-2x — Sb

= x+6 + x-2-2^x^-h-itx-12. (2)

Transposing and reducing,

-Vx^+2x~S5 = l--Vx'^-\-4:x-12. (3)

Squaring again,

af^^2x-S5 = l + x^+4:x-12- 2 V^Tl^^H^. (4)

Transposing and reducing,

Vx^-{-4:x-12 = x-\-12. (5)

Squaring again, x^ -i- 4: x — 12 = x^ -{- 24: x -{- 144. (6)

Hence a: =—7.8. (7)

IMAGINARY NUMBERS 77

This is the only solution obtainable, and yet it does not
satisfy the original equation. Hence that equation has no
true solution.

In fact, by substitution x= — 7.8 will be found to satisfy
equations (6), (5), and (4), but instead of satisfying

(3) it will be found to satisfy the companion equation
— Vrc^ -\-2x— 35 = 1 — ^x^ -\-AlX— 12, from which equation

(4) could equally well have been derived.

Further it is observed that a; = — 7.8 satisfies the equation

V^~+T + Va: — 5 = Va; + (3 H- ^x — 2,

which on squaring would make the sign of the radical in the
second member of (3) positive, and when this is squared it
will lead to (4).

Remark. In general, it is seen that by squaring, results are obtained
which belong to equations like the original except that the signs of some
of the terms are changed. In fact, squaring the equations x = a and
X = — a leads to the same equation, x^ = a?-. Roots which are introduced
by squaring are sometimes spoken of as " extraneous roots," but they are
really no roots at all of the original equation, and whether the solution
obtained is an actual root or not can only be determined by substitution.
The work should never be considered as finished until the character of
the solution has been ascertained.

114. EXAMPLES

Solve the following equations :

1. V3 a; - 7 = 25.

2. Vo- 2j;-3V2- 32: = 0.

3. V2 2; - 14 + V^' = V2 2' + 7.

4. V2 :r - 14 - V^ = V2.f + 7.
6. Vo a; - 1 + V2 2; + 5 = 12.

78

COLLEGE ALGEBRA

6. Va:2 - 32^ + 10 - V:r^ + 2 a: + 12 = 2.

7. Vb-10x-Vl-4:x=^5x-hU.

9. V:?; + 5 + V2 2: + 8 = 0.

10. Va; + 5 + V2 :z:2 _^ 7 2; - 15 = 0.

11. ^2 + 2: Va;2 +8a; + 40 = 2 + a;.

12. 2: + Va;2 + V5^2T^3^Tl = l.

13. x-\--\lx^-^dV5x^-dx + '3 = S.
6

14.

15.

V3^ + 5 V3^ + 1

V3^-l~ V8^-2'

5.

Suggestion. Regard as a proportion and apply addition and sub-
traction, by 42.

16.

^jV6x-h2 VV5a;-19
■\-V'bx—lc> yV5x + 13

V3 x — V5 X -j- 1

18.

19.

VI2 x-\-^Sx^-\-5x-{-lS ^^Q
Vl2^- V8a;''^ + 5:r+13

V2 a; + 5 + V3 a; + 10 V2 a; 4-~8 + V^V-h 26
V2 a; + 5 - V3a; + 10 ~ V2a:H-8 - V5 a: + 26

IMAGINARY NUMBERS 79

COMPLEX NUMBERS

115. In 87 numbers like V— 3, V— 5, etc., have been in-
troduced. Since V— 3 = V3 V— 1, V— 5=VoV— 1, and
in general V— ^ = VaV— 1, it is seen that all such numbers
can be expressed in terms of V— 1. The symbol i will be
used to denote V— 1; thus every pure imaginary number
can be expressed in the form of 5^, where 5 is a real number
and i the unit of pure imaginary numbers.

116. From the definition of i and from the theory of ex-
ponents we have for the powers of i the following :

zi =

h

v^ =

-1,

^3 =

-^^

i' =

1,

2*" =

1,

{in+1 ^

h

^•4n+2 _

-1,

^•4h4 3 _

— ^.

Thus the value of every integral power of i is contained
among + ^, — z, +1, — 1.

117. In the case of a complex number a + hi^ if a = 0 and
h=^0 the complex number becomes a pure imaginary number ;
if 6 = 0 and a ^ 0 it becomes a real number, and ii a = b = 0
it is equal to zero. If either a or h or both become infinite,
the number becomes infinite. It is seen that the complex
number is a twofold generalization of the elementary number
concept.

In making the extension consistency requires that the
extended number be subject to the fundamental laws of
algebra.

80 COLLEGE ALGEBRA

118. Definition. Two complex numbers, a + hi and a — hi,
differing only in the sign of the pure imaginary part, are
called conjugate complex numbers. It is seen that the con-
jugate of any complex number is obtained by changing i
into — i ; thus a + hi is the conjugate of a — hi and a — hi is
the conjugate of a+ hi. So also are hi and — hi conjugates.

119. Definition. The expression V^^ -\- ^2 taken with the
positive sign is called the modulus of a + hi, and the abbrevi-
ation mod will be used to denote "the modulus of." Thus
mod (a + iz) = Va^ + ^^. * If 5 = 0, that is, if the complex
number is wholly real the modulus reduces to Va^ which is
simply the absolute or numerical value of a.

It is seen that mod (^a + hi) = mod (a — hi'), since each is
equal to Va^ + h^.

120. Theorems. The sum, difference, product, and quotient
of two complex numhers is in general a complex number. For
let a-^ + h^i and a^ + h^i be two complex numbers, then their
sum, which is a-^-\- a^-\- (h^ + h^i is in general a complex
number. The difference, which is a-^^ — a^-\- (h^ — h^i, is like-
wise a complex number. The product is

(a-^a^ — h^^ + (^a^^ + ^^i)**

and is also a complex number. The quotient

^1 + ^1^* _ ^^1 + ^iO(^2 ~" ^2**) _ ^\^2 + ^1^2 I (^2^1 ~ '^1^2)^*
«2 + K^ ~~ (^2 + h^) C«2 - ^20 ~ <^2 + K ^i + ^2

is in general a complex number.

From this it follows that any rational integral function of
a complex variable is a complex number ; thus / (a -}- hi) is
of the form A 4- Bi, where A and B are real.

* The modulus of a number is also denoted by placing the number between
two vertical strokes. Thus the modulus of « + hi is denoted by |a + hi\.

IMAGINARY NUMBERS 81

It is also easily seen that if

then f (a — hi^ = A — B%

and therefore,

mod/ (a -f hi) = mod/ (a — hi) = -ylfi^a + hi) f(^a — hi).

SOME PROPERTIES OF CONJUGATE COMPLEX NUMBERS

121. The sum of tivo conjugate complex numbers is real and
equal to twice the sum of the real part of either.

Thus (a + hi) -\- (a — hi) =2 a.

122. The difference of two conjugate complex numhers is a
pure imaginary and is equal to twice the imaginary part of the
one which is used as minuend.

Thus (<^ + hi) — (a — hi) = 2 hi.

123. The product of two conjugate complex numhers is real
and positive and is equal to the square of their modulus.

Thus (a + hi) (a — hi) = a^ + V^.

The conjugate of f(a-\-hi) is f(a — hi)^ as is apparent
from 118.

IDENTITY THEOREMS FOR COMPLEX NUMBERS

124. If a + hi-= 0, then a = 0 and h = 0 ; for if not, we

would have , .

a = — 6^,

that is, a real number equal to a pure imaginary number,
which is impossible ; therefore a = 5 = 0, that is if a complex
numher vanishes^ hoth the real part and the pure imaginary
part must vanish together.

82 COLLEGE ALGEBRA

If a-^ + h-^i = a^ + b^i, then a-^ = a^ and h-j^ = h^;

for ^j — «2 + (^1 ~ ^2)*' ~ ^'
therefore, by 124, a^ — ^2 = ^i — ^2 = ^'
or a-^ = a^ and ^-^ = ^g,

that is, {f tivo complex numbers are equals the real fart of the
one must equal the real part of the other and the imaginary part
of the one equal the imaginary part of the other.

125. EXAMPLES

1. Find the sum of 2 + 3 i, V2 + V5 ^, VS — V2 2, 1 — 2 i.

2. Multiply 5 - 2 z by 3 + 4 «.

2 4- 5z

3. Find the value of ^—^

Q — I

4. Find the value of

V2 + V3z

5. Find the value of ( V2 — V3 iy.

6. Find the value of x

3 + 2 1 + 32

7. Find the value of 3 a;^ — 2 2 + 2 when 2 = 2 + 2.

8. Find the value of — — when z = 1 -\-2i.

GRAPHIC REPRESENTATION OF COMPLEX NUMBERS

126. In order to develop thoroughly the theory under-
lying the graphical representation of a complex number, it
is necessary to establish a number of fundamental principles.
For the sake of those who do not care to take up the details

IMAGINARY NUMBERS

83

of these principles the following customary treatment is
given :

If the line OM is assumed to have the length unity, the
student is already familiar with the method of representing
+ 5 by the point P, and — 5 by the point P\ which is at
the same distance from 0 in the opposite direction. But
— 5 = -}-5(— 1), and OP^ may be regarded as obtained by

Fig. 13.

revolving OP about 0 through an angle of 180° in the
direction indicated by the arrow head. That is, the factor
" — 1 " may be regarded as turning the line OP through
this angle of 180°.

In order to obtain a consistent method of representing the
quantity 5V— 1, we note that — 5 = 5 (V^^)^ or 5 z^.
That is, if + 5 is multiplied by i twice the result is — 5.
If the multiplication by i twice revolves the line through
180°, consistency demands that each multiplication by i must
revolve it through 90°, so that 5 V— 1, or 5 z, is represented

84 COLLEGE ALGEBRA

hj P" . Likewise, the representation of — 5V— 1, or 5 z^,
requires that the line be revolved through an angle of three
times 90°, or 270°. Hence — 5 V— 1, or — 5 i, is represented
by P'".

Since the ordinary complex number is the sum or differ-
ence of a real number and a pure imaginary, the method of
its representation must be analogous to that of the sum or
difference of two real numbers. Thus, to obtain that of
5 + 2, two units are measured to the right of P, giving Q ;
and that of 5 — 2 is two units to the left of P, or Q'. So
5 + 2 ^ is two units above P, or i^, and 5 — 2 ^ is two units
below P, or R' . The same results would be obtained by
first laying off the imaginary units from 0 upward or down-
ward, and then measuring five units to the right.

Addition or subtraction of complex numbers is accomplished
by a similar process. Thus, to obtain 5+2 ^ + (— 3 + 4 2),
three units are measured to the left of M and four are then
measured upward, giving S as the representation of the
sum. The same point aS' is obtained by taking the point T^
which represents the number — 3 + 4 z, and finding the fourth
vertex of the parallelogram of which T^ 0, and H are the
other three.

Since subtraction consists merely of the addition of the
corresponding negative number, it does not require a separate
discussion.

It is to be observed that in the above a complex number is
represented by a point whose abscissa is the real portion of
the complex number and whose ordinate is the coefficient
of i in the imaginary portion. That is, the complex number
is regarded as written in the form x + ii/.

By many authors it is considered a sufficient justification
of the whole method of representation merely to state that
since every point is represented by two numbers (coordinates)

IMAGINARY NUMBERS 85

and every complex number x + iy involves two independent
real numbers, every complex number may be represented by
a perfectly definite point in the plane, and conversely every
point in the plane represents a perfectly definite number.
Real numbers are given by points on the a:-axis, and pure
imaginaries by points on the j/-axis.

Note. Instead of regarding 5 + 2 i, —3 + 4 {, and their sum as
represented by R, T, and S, respectively, many authors regard them as
re2:)resented by the lines OR., OT, and O-S", respectively. In this form
the student of physics will recognize the parallelogram of forces and the
law of the combination of motion. The lengths of these lines are then
said to represent the moduli of these numbers, and the angles POR,
POT J and POS represent their amplitudes.

For the sake of those who desire a more scientific treat-
ment of these topics the following discussion is given, based
upon more fundamental principles.

127. If we take a point 0 in a straight line and lay off

OP
from 0 a positive unit OM. then the ratio of the sesr-

^ OM ""

ment OP, determined by a third point P in the line, to

the segment OM is some real number, positive or nega-.

tive according as P and 31 are

on the same or opposite sides

of 0. ^^x

In this case OP and 031
coincide in the same geometric ^^^ ^

line, though they may have
opposite directions. Moreover,

it is seen that to every point P in the line corresponds one
and only one real number, and conversely to every real
number corresponds one and only real point in the line.
Thus all real numbers, and only real numbers, have their
representation in the line.

86

COLLEGE ALGEBRA

Now let us consider the interpretation of

OP
OM

where OP

and OM do not lie in the same geometric line and where
account is taken of the angle from OM to OP as well as of

the lengths of the lines. This is
evidently a generalization of the pre-
ceding case, and it is apparent that
the ratio cannot be real unless OP
and OM are made to coincide again.
We have here an extension of
number, and in defining the laws
of operation which it shall obey, we
are only restricted to consistency
with previous definitions, such that when this number
becomes real no contradictions with previous definitions exist.

Definition. We define the number as determined when the
ratio of the lengths OP and OM is determined and when the
angle MOP is determined.

OP 0' P'

Thus ---- and —— — — define

Fig. 15.

OM ' O'M'

the same number if ratio of

the length of OP to OM is
equal to the ratio of the
length of O'P' to O'M', and
the angle MOP is equal to
the angle M'O'P'.

We adopt geometric addition,

that is OE-\-EQ=OQ,

OQ ^ OR
OM

Fig. 16.

and

RQ

OM^ OM'

and also

OR OM^OR
OM^ OQ OQ'

IMAGINARY NUMBERS

as a definition of multiplication. These are

evidently extensions of the same operations

when OR and OQ coincide, and all

of them are consistent with previous

definitions when the number

considered becomes real,

that is when OR and OQ lie

in the same line. ^^^' ^''

128. Let the line chosen in 127 be the axis of x.
Then OP

R

OM

= X. Erect a perpendicular OQ to OX 2ii 0, such

that the ratio of the lengths -p~ = y. Take S such that the

. OS

ratio of the leng'ths — is equal

^ OQ ^
to the lenecth ratio — ^ = y.

Then the number represented by

OS
M -TT^ is equal to that represented

-*^X OQ g^

by y—^ angles as well as ratios

of lengths being taken into
account, since both have the
same length ratio and both have
a positive right angle for angle.

OS

;Sh

T

Q

o

V

Fig. 18.

Hence

and

OS

OQ
OM

[0Q\'
\OMJ

OQ'

OS

OQ

OQ ^ OS
OM 031

But -~^ is real and negative and equal to — y^, since by con-

struction the length 0^ is a mean proportional between
that of O/S' and Oitf. Hence / QQY

88

COLLEGE ALGEBRA

Therefore

Similarly,
the length OQ.

OQ ^ .
OM ^'"

OM

= — yi if the length OQ' is equal to

Thus it is seen that all pure imaginary
numbers are represented by points in the axis of y.
In general, since

OP = OM' + M'P,

and

OP ^ OM' M'P
OM OM OM

we have

OP
OM

OM' OP'
OM OM'

X + yi.

where x is the abscissa of P and y is its ordinate.

129. Denote the number

have

qp_

OM

by z. We have seen
OP

(128) that ^ = X + yi.

Therefore z = x -\- yi. Let
OP = r. The ratio of

lengths

OP
OM

= r, and also

r = -Va^ 4- y^. The angle
MOP = cf> is called the am-
plitude or argument of the

X

Fig. 19.

z= x + yi= r (cos <f>-\- i sin (^) .

complex number. Snice -

T

^ y .
= cos (^, and - = sm <^, we

IMAGINARY NUMBERS

89

Geometrical interpretation of the four fundamental opera-
tions upon complex numbers can be given. Any two
complex numbers may be constructed with a common de-
nominator equal to the positive linear unit OM.

130. The addition of two complex numbers has already been
treated in the definition 127.

Thus if

and " - ^^2

2— -Q]^'

^1 + ^2- 031'^ OM

0I\ PJ\_OP^
OM'^ OM ~ OM

= z

3^

where P^P^ is drawn through Pj
parallel to OPg- Thus the numer-
ator of the sum is formed by com
pleting the parallelogram on OP^
and OP 2 as sides and taking the
diagonal from 0.
Since

OP2 + OP^ = OP 2 + P^P^ = OP3,

OP

.^OP,

OP.

_ ^^ 2

+

A^3

OP.

OM ' OM OM ' OM OM

or

^2 + ^1 = 2

3'

therefore

2i ~r~ Ziy — Ze) ~r" ^

1'

and the addition of complex numbers is commutative.

The difference of tivo complex numbers z^
structed geometrically. Let z.^ = x^ + y^^ and z^ = x^-\- y^i.

2j can be con-

90 COLLEGE ALGEBRA

OM OM'

^2 ~ ^1 —

Therefore

~ OM^ OM OM'

OPi , PiP. _ OP.
OM^ 0M~ OM'

qp^_qp^_pj\

OM OM OM '

Thus the numerator of the difference is constructed by
drawing a line from P^ to Pg*

131. The product of two complex numbers can be con-
structed as follows :

Let ., = -^ and ., = ^ = ^ • (Fig. 22)

jj. length OP3 length OPg mod (^2^1) mod z^

^^^^ length OPi ^ length OM' ^^ mod z^ ^ T^'

mod (^2^1) = mod z^ mod Zp
whence mod (z-^z^) = mod z^ mod z^.

Therefore mod (z2Zi) = mod (^z^z^~).

By construction, amplitude (^2^1) = amplitude
2| + amplitude z^, therefore amplitude (^z-^z^^
= amplitude z^ + amplitude z^

Therefore amplitude CH^-^^ = ampli-
tude (2^122)-

Therefore z^z,^ = z^z^ and the
multiplication is commutative, ^
though a different geometrical Fig. 22.

construction would be made for z-^z^.

IMAGINARY NUMBERS

91

OP2'

^2 ~ ^3^1-

mod z^ = mod z^ mod 2^,

Thus the product is formed by constructing the sum of
the amplitudes, and finding a line equal in length to the
product of the lengths of OP^ and OP2' Xhws,

0P^_ r

(Fig. 23)
132. In the case of the quotient of two
complex numbers^
let

therefore

Then
and

' or^i = ^,orr3=0Pi.0P2.

20 =

Fig. 23.

(131)

amplitude ^2 = amplitude z^ + amplitude z-^.

Therefore mod 23 = 2

mod

and

amplitude z^ = amplitude z^^
— amplitude z^.

Thus the amplitude of the dividend
minus the amplitude of the divisor is
to be constructed, and a line r is to
be obtained whose length r^ satisfies
the equation
OP

2 _ ^3

Fig. 24.

OM

= r.

3*

(Fig. 24)

133.

OP,

EXAMPLES

Simplify the following expressions and write the results,
(1) with positive exponents ; (2) in integral form ; (3) in
simplest form with radicals :

1.

-2 , 3

- 5 3

r-3/,4\¥

7-^ •»2

2.

i 1 . _2 . „ _.. _„

tn^n \ ^ fm ^n ^^ ^

-h ^

2 *
p^q-

92 COLLEGE ALGEBRA

3. x^x^ where a and h are the roots of the quadratic
equation y^ — 3 «/ + 2 = 0.

he

X{a-c){a-h)

5.

1 1

4.

X{a-b){a-c) . ^(6-c)(6-a)

1

X{c-a){b-c)
a b

6.

X{a-b)(a-c) . x(b-c)(b-a)

c

ab

Xib-cKa-b) . x^<^-('Kb-c)

X (c-a)(b-c)
1 1

7. x^x^ where a and b are the roots of the quadratic
equation y^ — 4 y + 4 = 0.

8. a^ -7-x^ where a and h are the roots of the quadratic
equation ^^ _ ^ _j_ 2 = 0.

8^(128)^ + V24 . VT5 • V2(3

8-^nys^'

</16^</2-</'6:l-</S2

,, V2 + V3 C"^"^^)^-

10. — — . 11^ „i m-ni-

2V 2 + V 3 Xm-n ' Va

IX-y Vj-^r ,-\V-^Z y~Z\y^x/

,2+X

13. (V2 + l)^. 15. (v'2-2V3)3.

14. (3-' - 2^ -}- 3^)2. 16. aJi9 + 8V3.

17. [15-4(14)2]i

19. ^V3 + V5-2\/15.

20. [(48)^+(162)^ + 12(6)^]i

IMAGINARY NUMBERS 98

Rationalize the denominators of the following fractions :
21. • -t- V-

22.

23.

24.

25.

V8

+ 2V2
5-3V2

V2

- V3 +

V5

x +

.2V^

^x

-v^

Va

+ V5-

V?

Va

-V3^ +

V^

yJx

+ 2V^-

-3V^

26

3 + V-2

V5-V-8

27.

7_5V-8
3+2V-5

28.

11 +z
7 — 5 ^ *

29

2

V2 — V3 i — V5 i

QH

^\fa + V^i — Vca

2 V^ + 3 V^ — -^x Vai — 'Vbi — Vc

Find the numerical value to 4 decimal places of the
following expressions, given V2 = 1.41421, Vo = 1.73205,
V5= 2.23607, V(j = 2.44949, V7 = 2.64575, and
V30 = 5.47723.

31. J ,. 33. 2-'^^^

2V7 4-V6 2V2-3V3

32. _2±V3_^ 3,_ 5

2V3-3V2 V2-V3 + V5

V2 - V3 + V6

35.

V2 + V3 + V6

Find the value of x in the following equations :

36. V^ + V2; — 3 + Vo; + 5 = 0.

37. 2V^^^+ V4a;-3- VlOa;+ll = 0.

94 COLLEGE ALGEBRA

38. + V8a:= V3a;-2.

39.

(jx + 5)^ -(x- ly ^ (^ - 2)^ - {x - 3)^

40. f^-=^Y+(:^^+3)^ = 1-^.

41.

42.

(flg + a:)^ _ (a — xy
a* + (a + a;)^ a^ - (a - o;)^

(2; + 2 a)* + (a: -2 a)* ^a

43. — ==— ' ^^=1: H ==^ = 4Va;^ — 1.

Va^ + 1 - Va;2 - 1 Vx'-^ + 1 + Va;2 _ 1

44. (a^ + x~^)^ = (a^ + a;)i

45. Vl3 + :i:"-^+ V2.r2 + 12= V2;2 + 29.

46. VS^HhTOI — V2^^^ = V2: — z. •

47

. Va: + 4 z — V:r = XJ 2 a: — .

2

48. Va^ — y^x -\- VI — a; = z.

Find the square root of :

49. 1-2V^T. 51. 4-6zV5.

60. 12^V6-6. 52. l-2zVl32.

CHAPTER VII

PROGRESSIONS

ARITHMETICAL PROGRESSION

134. Definitions. An arithmetical progression (A. P.) is a

8uccessio7i of teryns each of ivhlch is formed from the preceding
hy the addition (^algebraically^ of the same constant.

This constant is called the common difference of the terms.

It is obvious that each term may be formed from the fol-
lowing term by taking away the common difference.

The entire number of terms in an arithmetical progression
is unlimited. In what follows only a finite number of con-
secutive terms of the A. P. will be considered.

Let a denote the first term, d the common difference, I the
last term, n the number of terms, and s the sum of that por-
tion of the A. P. under immediate consideration.

It is obvious that the first term may be regarded as the
last and the last term as the first according to choice.

The ^th term of an A. P., considering a as the first, is

obviously a-\- Qn — 1) d. Considering the nth. term as the

last we have , ^ i^ -, x^ n

l = a+(^n-l}d. (1)

135. The sum of n terms may easily be found by writing
the terms in two forms. Thus,

s = a -f (a -f t7) + (a + 2 c?) -f- • • • + (a + 71—1 d) ;

s = l+(l--d}-hCl- 2d} +'■■-{- (I- /T^^l d).

95

96 COLLEGE ALGEBRA

Adding, we have 2 8 = na-\-nl,
or, s = |(« + 0. (2)

To find the sum of 80 terms of the series of even numbers
2, 4, 6, • • • . From (1) we have

Z= 2 + (80 -1)2 = 160.

and from (2) we have

s = -822-(2 + 160) = 6480.

It is to be observed that (1) and (2) are independent rehi-
tions between the five quantities a, Z, c?, n, s, so that, given
any three of them, we may find the other two. Thus given
a, c?, ^, to find I and s we have

I =z a + (^n — 1) d^
and substituting this value of I in (2) we have

n

s = -(^a -\- a + n — 1 ' tQ,

n

or, s = - (2 a + w — 1 • c?).

EXAMPLES

1. Find the common difference and sum of 30 terms of an
A. P., the first term of which is 6 and the hast term is 586.

2. The sum of a number of terms of an A. P. is 220, the
common difference is 2, and the last term is 31. Find the
first term and the number of terms.

3. The sum of 16 terms of an A. P. is 256, and the last
term is 31. Find the first term and the common difference.

4. The sum of 5 terms of an arithmetical progression is
35. Find the third tei'm.

Suggestion. Let the series be x — 2d, x — d, x, x -\- d, x -\- 2d.

PROGRESSIONS 97

136. Definitions. Let a, A^ h be three consecutive numbers
in arithmetical progression. Then A is called the arithmet-
ieal mean between a and h.

By definition, A-a = h-A,

or, 2 J. = a + 5,

A = --.

In the same way if a, A^ B, «7, • • •, 5 are consecutive terms
of an A. P., then ^, J5, (7, • • • aie said to be arithmetical meayis
between a and b.

137. The problem to insert ru arithmetical means between

p and q may be readily solved, for we have the first term

a—f^ the last term I = g, and the number of terms n — m-\-2,

so that 7

7 I — a Q — P

d = -, or, ^— ^.

^i — 1 m + 1

Therefore the terms are

m + 1 wi + 1 7n-\-l

EXAMPLES

1. Insert 5 arithmetical means between 2 and 16.

2. Insert 7 arithmetical means between — 3 and 21.

3. The sum of 8 numbers in A. P. is 9 and their product
is 15. Find the numbers.

Suggestion. Let x — d, x, x -\- d be the numbers.

4. If the 5th term of an A. P. is 7 and the 12th term is
50, find the 20th term.

5. If the 14th term of an A. P. is — 2 and the 19th term
is 16, find the 21st term.

98 X COLLEGE ALGEBRA

6. If the 20th term of an A. P. is — 15 and the 4th term
is 4, find the 12th.

7. The population of a certain town increases annually in
A. P. In 1892 it was 1023; in 1900 it was 1175. What
was its population in 1888 ? What is it in 1907 ? What will
it be in 1921?

8. It has been shown in mechanics that, if the resistance
of the atmosphere be neglected, a falling body moves over a
distance of 16.1 ft. in the first second and 48.3 ft. in the sec-
ond second, and so on in arit/.imetical progression. How far
will it fall in 10 seconds ?

9. A body falls ^g ft. in the first second and | g ft. in the
second and so on in A. P. How far does it fall in t seconds?

Note. The law expressed in example 9 still holds when t is not an
integer, but the proof of this is obtained by means of the calculus.

10. Assuming the conditions of example 8, how long would
it take the body to fall 788.9 ft.?

11. A farmer is to build a fence 825 ft. long with the posts
one rod apart. He must carry these one at a time from one
end of the line. How far will he have traveled when the
posts are all distributed and he has returned to the starting
point?

12. The three digits of a number are in A. P., their sum
is 12, and the last digit exceeds the first by the amount of
the middle digit. Find the number.

13. A number consists of three digits in A. P. Their sum
is 15 and their product is 105. Find the number.

14. A number consists of 4 digits in A. P. Their sum is
20. The product of the first and fourth is to the product of
the second and third as 2 is to 3. Find the number.

PKOGRESSIONS 99

15. If «^ h\ c^ are in A. P., then , , are in

.p b -\- c c ■\- a a-\-o

16. Show that the sum of any 2 7i + l consecutive integers
is divisible by 2/1 + 1.

17. Show that the sum of any 2 n consecutive integers is
not divisible by 2 n.

18. The sum of any number of consecutive odd integers be-
ginning with 1, i.e. 1, 3, 5, • • • is a perfect square.

19. If 2n-\-l terms of the series of the last example be
taken, then the sum of the alternate terms 1, 5, 9, • • • is to
the sum of the remaining terms 3, 7, 11, • • ■ 'ds n -{- 1 is to n.

GEOMETRICAL PROGRESSION

138. Definition. A geometrical progression (G. P.) is a suc-
cession of terms such that the ratio of any one to the preceding
is constant throughout the progression.

It is thus seen that each term is formed from the preced-
ing by multiplying by this constant ratio.

Example. Show that the terms of a G.P. form a con-
tinued proportion.

Here, as in arithmetical progression, the entire number of
terms is unlimited.

Let a denote the first term, r the constant ratio, I the last
term, s the sum of the terms, and n the number of terms of
that portion of the progression under consideration.

The nth. term of the progression considering a as the first

term is obviously ar""i. Considering the nxh. term as the

last,

l=ar^-^. (1)

100 COLLEGE ALGEBRA

139. The sum of n terms can readily be found as follows :

s= a-i- ar -f ar^ H- • • • + ar""^ -\- ar'^~^^
rs = ar + ar^ + ar^ + • • • + ^r"~i + ar",

or subtracting we have,

s — sr= a — ar^^

or s(l — r)= a(l— r'^),

or «=— ^ -' (2)

1 — r

This formula for the sum may also be obtained as follows:

s = a + ar + ar^ + • • • + ar^~^
= a(l + rH-r2+ ■•• + r"-i)

^'i^l^n. (106)

1 ~ r

Example : To find the sum of 6 terms of the G.P., 1, 2,

4, •••wehave .-, ^ex

8=1^=^ = 63.
1-2

It is to be observed that (1) and (2) are independent
relations, and that given any three of the five quantities
a, Z, r, 72, s, we can determine the other two.

Thus, given a, Z, w, to find r and s. From (1) we have

^ a

and from (2) we have _ 7""i[^

s= — •

PROGRESSIONS 101

EXAMPLES

1. The first term of a G.P. is 2, the ratio is 3. Find the
fifth term.

2. If the first term of a G.P. is 2, the last term 162, and
the ratio 3, find the sum of the terms.

140. If a, (7, h are three consecutive terms in a G.P.,

then G- is called the geometrical mean between a and h.

Since by definition ^ ,

^— A
a (r'

a'^=ah,
or ' G- = ± ^ah.

If a, (r, H^ /, •••,5 are consecutive terms in a G.P., then
G-^ H^ I^ ••• are called geometrical means between a and h.

The problem of finding m geometrical means between a
and h is reduced to that of finding the constant ratio, and
this has already been done in 139.

Example. Insert 5 geometrical means between 2 and 128.
Here the number of terms is 7, so that

128 = 2 . r6 or r6 = 64 and r = ±2.

141. EXAMPLES

1. Find the sum of six terms of the G. P. 3, — 2, |, •••.

2. Find the fifth term and the sum of five terms of the
G P 12 9 -2_"L ...

3. Insert 5 geometrical means between 2 and 3.

4. Given I = 1215, a = b, n=Q, find r and s.

5. Given a = 21, 1 = 5103, r = 3, find n and s.

6. Given ?^ = 5, r = ^, ^ = 2, find a and s.

102 COLLEGE ALGEBRA

7. Given n = 4, r=— 2, ? = 25, find a and s.

^ . 7 8r — s -\- a „ sr — s-\-a Ir

8. Given a, r, s, prove t = ■ — , r" = = — •

r a a

9. Given a=2, r= — |, s = -y^-, find I and ti.

10. The sum of three numbers in G. P. is 21, and their
product is 216. Find the numbers.

Suggestion. Let -, x, xy, represent the numbers.

y

11. The sum of three numbers in G. P. is 7 and the sum
of their squares is 91. Find the numbers.

12. There are four numbers of which the first three are
in G. P. and the last three in A. P. Their sum is 13, and the
third minus the first is 3. Find the numbers.

13. If ^, ^, c, d are in A. P., prove that

(a2 + y^ + c^)(hc + hd^ ed) = (P + c'^ + d?){ah + hc^ ca).

14. Three numbers whose sum is 15 are in A. P. and if
1, 4, 19 be added to them respectively the results are in G. P.
Find the numbers.

142. If the ratio is less than unity the sum of n terms
remains finite when ?z = co. Considering the formula for

the sum, we have „

s =

or

1 — r
a ar^

1 — r 1 — r
ar''

in which may be made as small as we please by taking

n sufficiently great, since r is less than unity.
We have therefore

s = when n is infinite.

1 — r

PROGRESSIONS 103

EXAMPLES

1. Find the sum of 1 + J + ;| + ••• to infinity.

2. Find the sum of -^q + yl^ + j-^q-q + •••to infinity.

3. Find the value of 0.29.
Suggestion. 0.29 = j%% + j-^'^ + .-.

4. Find the value of 0.325. 5. Find the value of 2.21.

HARMONICAL PROGRESSION

143. Definitions. A succession of terms the reciprocals of
which are consecutive terms of an arithmetical progression is
called a harmonical progression (H. P.).

If «, H^ h are three consecutive terms of a harmonical progres-
sion, then H is called the harmonical mean between a and h.

By definition -, — , - are three consecutive terms of an
a H h

1 l^\

A. P. so that == — - — ,

xz 2

rr 2 ah
or H=

a + 5

Again if a, H, K, L^ -"^h are consecutive terms of a har-
monical progression, then H^ K^ L^ ••• are called harmonical
means between a and h.

144. The arithmetical, geometrical, and harmonical means
between two numbers a and h have been denoted by A^ (r, H^
respectively, and the relations

2 ■:-

a=^±^ab.

104 COLLEGE ALGEBRA

have been obtained. From these it appears that

or a = ± VAR.

That is, the geometrical mean between two numbers is the
geometrical mean between the arithmetical and harmonical
means of the same two numbers.

From the relation G = VAII it appears that (5^ lies between
A and R.

145. To show the connection between the harmonic pro-
gression of algebra and the harmonic division *. of a line in
geometry, let the line AB be divided harmonically (in the
geometrical sense) at P and P^ Denote AP^ AB^ and AP'
by x^ ?/, and z respectively. Then from the harmonic divi-
sion of the line we have

Fig. 25.

or

or

or

^-^_

^-y

X

z

X z

y =

2xz

x-\-z''

AB =

2AP-

AP'

AP^-AP'

* Problems in H. P. may be solved by reducing them to the corresponding
problems in A. P., but the general foiTuula for the sum of an H. P. has not
been found.

PROGRESSIONS 105

That is, when a line AB is divided harmonically at P and
P', AB is the algebraic harmonic mean between AP and
AP', or AP^ AB, and AP' are in harmonical progression ;
and conversely if AP, AB, and AP' are in harmonical
progression, AB is divided harmonically at P and P' .
This shows that harmonic progression of algebra is identical
with harmonic division of geometry.

146. EXAMPLES

1. Find the tenth term of the harmonic series, J, -|, J, ••••

2. Insert five harmonic means between 3 and 15.

3. If A, G-, and H denote the arithmetic, geometric, and
harmonic means between two positive numbers a and h,
arrange these three means in their order of magnitude.

4. Find the harmonic progression of which the eleventh
term is \, and the fifteenth term is \.

5. Find the H. P. if the thirteenth term is \ and the
nineteenth term is \.

6. Find the seventh term of the harmonic progression

1-4- 15. 5 ...

7. If 2 is a harmonic mean between a and h, then

1 +^ = Ui

z — a z — b a h

a

8. If a, h, and c are in H. P. show that

. TTT^ h -\- c c -\- a a-\-h

are m H. P.

9. If a, h, and c are in H. P., then a, a — c and a — h are
in H. P. So are c, c — a, c—h also in H. P.

106 COLLEGE ALGEBRA

147. EXAMPLES

1. Find the 12th term of V2, — 2, 2V2, — 4, etc.

2. On the ground are phiced n stones ; the distance be-
tween the first and second is 1 yd., between the second and
third 3 yd., between the third and fourth 5 yd., etc. How
far would a person have to travel to bring them one by one
to a basket placed at the first stone ?

3. Find the sum of 15 terms of 3, 7, 11, •••.

4. Find the sum of 19 terms of 1, 1|, 2, 2|-, ••-.

5. Find the sum of 7 terms of J, \^ \^ •••.

6. Find the sum of 10 terms of -^g^, — |, -|, •••.

7. Find the sum to infinity of |, ||, |^f, •••.

8. Find the sum to infinity of — |, \^ — |, •••.

9. Find the value of 0.127.

10. Find the value of 0.245307.

11. Given the sum of the series 21, 19, 17, etc., to n terms
to be 120. Find the last term and the value of n,

12. Divide unity into four parts in A. P. such that the
sum of their cubes is -^^.

13. Find an A. P. such that the sum of the first five terms
is one fourth of the sum of the following five terms, the sum
of the first ten terms being unity.

14. Find the series of arithmetical means between 1 and
21 such that their sum has to the sum of the two greatest
of them the ratio 11 : 4.

15. A number of three digits in A. P. is equal to 26 times
the sum of its digits. If 396 be added to the number, the
digits are reversed. Find the number.

PROGRESSIONS 107

16. Snm to infinity V^ + l^ L_, 1, ....

V2 - 1 2 - V2 2

17. Show that the square root of 0.444 is 0.666.

18. In a G. P. show that the product of any two terms
equidistant from a given term is always the same.

19. In a G. P. if each term be subtracted from the suc-
ceeding term, tlie differences are also in G. P.

20. The square of the arithmetical mean of two quanti-
ties is equal to the arithmetical mean of the arithmetical
and geometrical means of the squares of the same two
quantities.

21. Prove that the two quantities between which A is the
arithmetical and Gr the geometrical means are given by the
formulas ^ ^ ^/CA+ a)(A~ G).

22. Suppose a body to move 20 miles the first minute,
19 miles the second, 18 miles the third, etc. How far will
it go before it stops ?

23. Three quantities are in H. P. If -|- the middle term
be subtracted from each, show that the three remainders are
in G. P.

24. Show that h^ is greater than, equal to, or less than ac
according as a, 5, e are in A. P., G. P., or H. P.

CHAPTER VIII
PERMUTATIONS AND COMBINATIONS

148. Definition. A permutation is a group of some or all
of any number of objects taken in a definite order ^

149. Definition. A combination is a group of some or all
of any number of objects taken without reference to their order.

Definite arrangement characterizes permutations, while a
mere selection regardless of definite arrangement character-
izes a combination.

Thus, four students on a seat, taken as a group, form but
one combination : but they could evidently be arranged in
many ways, each of which would be a permutation. Also
any three of the four, if taken as a group, would form a
single combination but several permutations.

150. Notation. We shall denote the number of permutations
of n different objects taken r at a time by „P,,, and the number
of combinations of n different objects take7i r at a time by „(7^.

Other notations for ^P^ are "P,., P„^^, and for „6',., "^OJ.,
O

151. Fundamental Principle. In the treatment of permu-
tations and combinations the following fundamental principle
is used : If one operation can be performed in m different ways^
and after that a second operation can be performed in n differ-
ent ways for each way in which the first has been performed^ the
number of different ways in which the tivo operations can be

108

PERMUTATIONS AND COMBINATIONS 109

performed together is mn. For with the first way of per-
forming the first operation there are n different ways of per-
forming the second and thus n different ways of performing
the two together ; with the second way of performing the
first, there are n different ways of performing the second,
and thus n other different ways of performing the two
together. Continuing in this way, it is seen that the whole
number of different ways of performing the two operations
together is n -\- n -f • - -f ?i to m terms, or 7nn. In the same
way if an operation can be performed in m different ways,
and after that a second in n different ways, and after that a
third in p different ways, the number of different ways of
performing all three together is mnp. For it has just been
shown that the number of different ways of performing the
first two together is mti^ and hence as before the number of
different ways of performing these two and the third is mnp.
And in general if an operation can be performed in m^ dif-
ferent ways and afterwards a second in m^ different ways, ••
up to an nth in m,^ dift'erent ways, then all can be performed
together in m^m^ •-• m„ different ways.

Thus if a traveler can go from Syracuse to New York in

3 different ways, and then from Ncay York to Boston in 4
different ways, the number of different ways in which he
can go from Syracuse to Boston via New York is 12.

PERMUTATIONS

152. To find the number of permutations of four students
taken three at a time., or^ what is the same thing., to find the
number of different arrangements iyi which they can be seated
three at a time in a line.

Of the 3 seats at our disposal any one of the students can
occupy the first. Therefore the first seat can be filled in

4 ways. After the first seat has been filled, any one of the

110 COLLEGE ALGEBRA

remaining 3 students can occupy the second for each way
in which the first has been filled, and therefore the two
together can be filled in 4 x 3, or 12, ways. After the first
two seats have been filled, either of the remaining two students
can occupy the third, and therefore the 3 seats together can
be filled in 4 x 3 x 2, or 24, ways.

153. We are now prepared to state the general theorem :
The number of permutations of n different objects taken r at a
time {^P,) is ^(^^_i)(^^_2)(^_3)...(,,_^_l_l).

Proof. We may represent the objects by the n letters
«j, a^, ^3, •••, a„. This problem is the same as to find the
number of different ways in which the n letters can be
distributed r at a time in a box with r compartments, one
letter in each compartment. We can place any one of the
n letters in the first compartment, thus filling it in n
ways. After the first compartment has been filled we can
place any one of the remaining n — \ letters in the second
compartment for each way in which the first has been filled.
Therefore we can fill the two together in n (n — 1) different
ways. After the first two have been filled we can place any
one of the remaining n -• 2 letters in the third compartment
for each way in which the first two have been filled. There-
fore we can fill the three together in n Qn — 1) (ji — 2)
different ways.

Reasoning in this way, and noticing that for each com-
partment filled a new factor is introduced which is one less
than the preceding factor, and that the number of factors
in the product is always equal to the number of compart-
ments filled, we obtain as the last factor, when r compart-
ments are to be filled, / i\ ,1

' n — (r — 1) ov n — r -[- 1,

Therefore „P^ =n(n- r)(n - 2)(7i - 3) ... (w - r + 1).

PERMUTATIONS AND COMBINATIONS 111

154. Corollary. If all the objects are taJce7i together^ r = n^
and the last expression becomes

„P, = n(n-l)(n-2)...3.2.1.

This is the continued product obtained by taking all the
natural numbers from 1 up to n, inclusive, and is called
factorial n,^ and is denoted by the symbol 7il or \n. We
sh-all employ the former and write

155. EXAMPLES

1. In how many different ways can six people walk abreast ?

Beginning at one end of the line any one of the six can
occupy the first place, and any one of the remaining five the
second place ; thus the first and second places can together

* The definition of factorial n as given above applies only when n
is a positive integer. It, however, is entirely consistent with the two

equations (w — 1) ! = — , and 2 ! = 2, since n ! = n • (n — 1) \. We shall

71

use these two equations as a general definition of factorial n, applying
it to all integral values of n, whether positive, negative, or zero. The
second equation is necessary, since the first defines only the ratio between
n! and (n — 1)!, and some other statement is necessary in order to give
the absolute value of n !. Starting with 2 1 = 2, the form n ! = n • (n — 1) !
will enable us to build up the factorials of all positive integers ; while
the form (n — 1) ! = — will give the factorials for the lower values

of n. Thus

1! = 2] = 1,

9 '

0! = li = l,

1

(-l)!=:^=oo,etc.,

and by continuing the process we see that the factorial of every negative
integer is infinity.

112 COLLEGE ALGEBRA

be occupied in 6 x 5 ways ; by continuing this reasoning we
see that the whole number of ways in which the six people
can walk abreast is gPg =6x5x4x3x2x1 = 720.

2. Given eleven trees. In how many different ways can
a row of three of them be set out ?

Any one of the eleven can be set in the first place, and
afterwards any one of the remaining ten in the second, and
after that any one of the remaining nine in the third,
and thus the row of three trees can be set in 11 x 10 x 9 = 990
different ways.

3. In how many ways can eight numbered seats about a
round table be occupied by eight people ?

Here the emphasis is placed, not upon the relative posi-
tions of the persons, but upon the different occupation of
the seats by the persons. Thus the first seat can be occu-
pied by any one of the 8 persons, the second by any one of
the remaining 7, and hence by continuing the process we
obtain gPg as the number.

4. In how many ways can eight people be seated with
respect to each other about a round table ?

Here the emphasis is not upon the different occupation
of the seats, but upon the relative positions of the persons.
Therefore the first person may take any place without affect-
ing the arrangement ; after that the next person may take
any one of 7 seats, the next any one of 6, and so on, and
thus the number of ways in which 8 people can be seated
with respect to each other is ^P^-

5. In how many ways can eight numbered seats around a
table be occupied by four gentleman and four ladies, so that
no two gentlemen sit side by side ?

The first can be occupied by any one of the 8 persons,
but the second is now restricted to one of 4, the next is re-

PERMUTATIONS AND COMBINATIONS 113

stricted to one of 3, and so on, giving 8x4x3x3x2x2
X 1 X 1 = 8 X 4P4 X 3P3 = 1152.

6. In how many ways can four gentlemen and four ladies
be seated with respect to each other about a round table so
that no two gentlemen sit together ?

One seat may be occupied without affecting the order of

arrangement. The rest can be occupied in ^P^ x 3P3 = 144

ways.

COMBINATIONS

156. Let us consider the number of combinations of 5
students taken 3 at a time. 5C3 is the required number of
combinations. We are to find its value. Each combination
contains 3 students, and therefore will yield 3P3 = 6 permu-
tations, and therefore ^6^3 combinations will yield gC'g x 6
permutations. But this is the whole number of permuta-
tions of 5 things taken 3 at a time, and is equal to 5P3, or 60.

Hence 6X5(73=60.

Therefore 5 Cg = 10.

157. Let us now take up the general theorem of which the
foregoing is a special example.

Theorem. The number of comhinations of n different objects
taken r at a time, or ^^C , is equal to the number

7i(n—l)(n — 2) ■■• (n — r-\-l)
1.2.3--.r
Peoof. „C^ is the required number of combinations.
Each of the „{7,. combinations contains r objects, and there-
fore will yield j^P^ = r\ permutations, by 154. Hence „(7^
combinations will yield „(7^ x r ! permutations, and as this is
the whole number of permutations of n different objects
taken r at a time, or „P,,, we have

nt,. X rl = nPr-

I

114

COLLEGE ALGEBRA

Whence, dividing by r !,

n _ n^r _ ^(^-l)(^-2) .♦• (n-r+1) .- „.

/* .

Corollary 1.
For, „(7,=

and

1.2.3...r

P — P

T

n(n—V)"- {n — r-{- 2)'
. 1.2--.(r-l) _

(n — r-\-X)

n _ 92 (n — 1) • • • (y^ — r + 2)
" '-i~ 1.2...rr-n

Corollary 2.

For, _a = ^^^±^

n+1 '^r — • n^r—Y

n+1 ^r

n (?2 — 1) • • • (92 — r + 2)"
(r-1) ...2.1 ^.

158. Multiplying both numerator and denominator of
the expression found for ^Or in 157 by (?2 — r)! = (/i — r)
{n — r — X) ••.3.2.1, we have

r\{iii — r)\ '

where the numerator is now seen to be the continued product
of all the integers from 7i to 1, or yi !,

/Ml

and therefore

n ___JL___
T\{n — r)\

As between this formula for „(7^ and that of 157, the latter
is to be preferred for simply calculating „C^^, though an
equivalent formula which is given in 161 is sometimes to be
preferred to either, while the formula of this section is more
compact and in many relations is to be preferred.

PERMUTATIONS AND COMBINATIONS 115

159. Thus far the symbol „0q has no meaning, but we may
define it as the value which the above expression for „C^
becomes when r = 0, that is

n^o = 77T-H- = 1- (154, footnote)

160. EXAMPLES

1. In how many ways can the' seats of a four-oared shell
be occupied by 8 candidates ?

2. In how many ways can the crew, as a whole, be chosen
in the preceding example ?

It is to be noted that the first examplie concerns the
arrangement of the crew in their seats, the second only the
selection of the crew.

3. In how many ways can a committee consisting of 3
freshmen and 4 sophomores be chosen from 7 freshmen and
8 sophomores ?

The 3 freshmen can be chosen in ^ C^ ways ; the 4 sopho-
mores in g6^4 ways ; each choice of 3 freshmen can be com-
bined with each choice of 4 sophomores to form a committee ;
therefore the whole number of possible committees is

.a x^a = — — X ^•'^•^•^ = 2450.

^ ^ ^ * 1-2.3 1.2.3.4

4. From 7 different roses, 6 different carnations, and 5
different chrysanthemums, in how many ways can 2 roses,
3 carnations, and 1 chrysanthemum be chosen? A71S. 2100.

161. Theorem. The nu7nber of combinations of n different
objects taken r at a time is equal to the yiiimber of combinations
of the n objects taken n — r at a time^ orJJr = n^n-r- For
every time we select a group of r things, there remains a
corresponding group of n — r things. Therefore the number

116 COLLEGE ALGEBRA

of groups of n different things taken r at a time is equal to
the number of groups of the n things taken 7t — r at a time.
This may also be proved as follows :

n _ n\

r ! ( >t — r) !

n _ "^ • _ n\

(/i — r) ! [n — (r^ — r)J ! {n — r)\ r\

and therefore nOr = n ^n-v

When n objects are divided into two groups of r and n — r
respectively, the groups so formed are called complementary
combinations.

The relation is of great convenience in practice. Thus to
find the value of 53650 by the formula given in 157 would be
a tedious process, but instead we use its equal,

,,(7,= ^^- ^^-^^ = 23426.

162. Theorem. n+l(^r = n^^r + n^r-V

Pkoof. We have „C^,. = „C^_i • ^~ • (157, Cor. 1)

r

Whence, by addition,

= »^.-i • {^) = ...1^. (157, Cor. 2)

Another Proof. Let us set aside for the moment one of
the n-\-l objects. Then the number of combinations of the
n -\-l objects r at a time which do not contain this object is
JJj. ; and every combination containing it must contain r — 1

TERMUTATIOXS AND COMBINATIONS 117

of the other n objects, and so there will be n^r-i of them.
But the two classes together make up the whole number of
combinations of the n -\-l objects taken r at a time, and
therefore n _ n ^ n

The latter method of proof leads to a more general theorem
of which the above is a special case. If r, s, m, and 7i are
positive integers, such that r -\- s = ???, then

For, suppose the m objects to be divided into two groups
of r and s objects respectively. Then „j(7„ includes the
combinations for which the entire 7i come from the first
group, those for which 7i— 1 come from that group, and one
from the second group, and so on until finally it includes
those for which all come from the second group. That is,

it includes ^C„, r^n-i ' s^v ^"^^ ^'^ ^^^ ^^ s^n- If either r
or s is less than n^ certain terms in the development will
vanish.

If in this development r, ?^, and s become w, r, and 1,
respectively, the theorem n+i^r = n^r-^ n-i^r i^ obtained.

163. Let us now examine the series

P n P ... O ... P

„v^Q, n\-^Y> 71^2' ' n^r'i "> n^n'

»^. = ''~l^'^ ■ nO,.-v (157, Cor. 1)

For small values of ?% is in general greater than

1, but as r increases the numerator will decrease and the

denominator will increase, and thus will decrease,

r

and eventually will become less than 1. So long as it is
greater than 1, „(7^ will be greater than jfir-v ^^^^ ^^^^ ^^^~

118 COLLEGE ALGEBRA

cessive terms of the series will be growing larger, but after
becomes less than 1 they will be growing smaller.

Let us find the value of r for which ^O^ is greatest.

We observe that there are n + 1 terms in the series, and
there are two cases :

First when n is odd^ and hence n-\-\ even. In accordance
with the preceding and 161 they can then be arranged in
pairs of equal numbers from below upwards in order of
ascending magnitude as follows :

O — C

C — O
fi — n

Since there are r + 1 in each column, and n + 1 all together,
we have o^ . i\ . i

ryi T

therefore r =

and n — r =

2

71 + 1

2

Therefore when n is odd there are two terms, n^n-\ ^-nd

n^n+v which are equal to each other and greater than any

other members of the series.

Second^ when n is even^ and hence ?^ -f- 1 odd. In this case,
since ^ -f- 1 is odd, the greatest term of the series is left
unpaired, and we have the following arrangement ;

PERMUTATIONS AND COMBINATIONS 119

C

C — P

o — o

C — O

n^O — n ^n*

Since there are r pairs,

we have 2 r + 1 = ?i + 1,

n

and J^n is the greatest term of the series.

2

164. Theorem. Tlie number of ways of dividing m + n
different objects into tivo groiqjs of m and /^, respectively^ is
(ni + 7i) !
m\ n\

Suggestion. This is equivalent to selecting m objects
for the first group. See 158. Let the student supply the proof.

Theorem. The number of ways of dividing m -^ n -{- p differ-
ent objects into three grou2:>s of m, n^ and p^ respectively, is
(m -\- n+ p)\

ml n\ p\

Proof. The group of m objects can be selected from the
m-\-n-\- p in m+n+p(^m ways ; and for each way in which they
are selected the group of n objects can be chosen from the
remaining n-\-p in n+p^n ways and therefore the two selec-
tions can be made in

n V p _<J^i-^n-\- py. (n-\-py. (m + n-{-py.

m\{n+ p)\ n\ p\ m\ n\ pi

ways.

120 COLLEGE ALGEBRA

165. Theorem. The number of j^armutations of n objects, k
of which are of one hind, I of another kind^ and m of a third
kind, and the rest all different, is equal to

nl
k\ l\ m\

Proof. Let x represent the required number. If the k
like things be replaced by k unlike things, each of the x per-
mutations will yield k ! permutations, and therefore x of
them will yield x - k\ permutations. If next in each of the
X ' k\ permutations the I like things be replaced by I un-
like things, we shall similarly obtain x • k\ l\ permutations.
Finally, if in each of these the m like things be replaced by
m which are unlike, we shall have x - k\l\ m\ permutations
of n differeyit things taken all together. But this number is
equal to n ! by 154.
Therefore X'k\l\m\=n\,

^^^ ^^TTT} — V

kl 1 1 ml

166. Theorem. The number of permutations of n different
things taken r at a time, when any one of them can occur once,
twice, and so on, up to r times in each permutation, is equal to n^.

Proof. The first place may be filled in n ways, and the
second can also be filled in n ways, since each thing may be
used more than once, and hence the two together can be filled
in nxn = n^ ways ; in the same way each place may be filled
in n ways, and therefore the number of ways in which the r
places may be together filled is nxnxn ■•• to r factors, or w'".

167. Theorem. The total number of combinations of n things
taken one at a time, two at a time, a7id so on up to n at a time,
that is, /> I /^ I /H' I I n

n^l~r n^2~^ n^3~r '" -Tn^n''

is equal to 2" — 1.

PERMUTATIONS AND COMBINATIONS 121

Proof. This total number of combinations includes the
number of all possible ways in which a selection may be
made by taking some or all of the things at a time. Hence
in making the selection each of the objects can be treated in
either of two ways : it can be chosen or rejected. Hence
the total number of ways of treating all the objects is
2 X 2 X •••to n factors, or 2" ways. But this includes the
case in which all the objects are rejected, and the number
of actual selections, or

is equal to 2" — 1.

Another proof of this is given in 181, Cor.

168. EXAMPLES

1. How many different selections of 5 coins can be made
from a bag containing an eagle, a dollar, a dime, a sovereign,
a shilling, a mark, a franc, a lire, a gulden, a krone, a cent,
a pfennig, and a farthing ?

2. How many numbers can be made using all of the
digits 3, 5, 7, 8 ?

3. How many numbers over 6000 can be made with the
digits 3, 5, 7, 8 ?

4. How many different arrangements can be made with
the letters of the word decimal when the vowels occupy the
even places ?

5. How many numbers of 8 figures can be made with
the digits 1, 2, 3, 4, 5, 6, 7, 8 when the odd and even digits
alternate ?

6. How many numbers of 8 figures can be made with the
digits 0, 1, 2, 3, 4, 5, 6, 7 when the odd and even digits
alternate, zero being considered as even ?

122 COLLEGE ALGEBRA

7. If the number of permutations of n things taken 4 at
a time is 20 times the number of permutations of ti — 2
things taken 3 at a time, find n,

8. From 10 classical and 8 philosophical students, how
many different committees can be formed each containing

4 classical and 3 philosophical students ?

9. If 15(7^ = igO;.^, find ^(78, and ^^.^r-v

Note. — In problems 10, 11, 15 a word is understood to mean a suc-
cession of letters.

10. Out of the 26 letters of the alphabet, in how many
ways can a word be made consisting of 5 different letters,
2 of which must be a and e ?

11. How many words can be formed by taking 3 consonants
and 2 vowels from an alphabet containing 21 consonants and

5 vowels ?

12. A stage will accommodate 5 passengers on each side ;
in how many ways can 10 persons take their seats when 2 of
them remain always upon one side and a third upon the
other ?

13. How many different arrangements can be made from
all the letters of the words Mississippi ? Cincinnati ? Phila-
delphia ?

14. How many different numbers of 8 figures can be
formed with the digits 2, 2, 3, 3, 3, 5, 7, 7 ? How many of
9 figures with the digits 1, 1, 4, 4, 4, 0, 0, 6, 6 ?

15. How many words can be formed from the letters of
the word Onondaga, so that the vowels and consonants occur
alternately in each word ?

16. A war vessel has a signaling system of five colored
electric lights ; each color has 3 distinct positions. Find the
total number of signals that can be used.

PERMUTATIONS AND COMBINATIONS 123

17. In how many ways can n things be given to p persons
when there is no restriction as to the number of things each
may receive ?

18. How many numbers of 4 figures each can be formed
with the digits 2, 3, 5, 7, 8, 9 when there is no restriction as
to the number of times a digit may be repeated in each
number ?

19. In how many ways may a sum of money be drawn
from the bag mentioned in example 1 ?

CHAPTER IX

BINOMIAL THEOREM

169. In this chapter we shall obtain a development for
the 7ith power (when n is ?i positive integer) of a binomial
a-\- b, and show that,

+ „(7X-^5^'+ ...+,(7,5^ (1)

By actual multiplication we obtain :
(a-\-hy = a^-\-2ah-\-b^ = ^CQa:^ + ^C^ah + ^aj)% (159, 157)

(a + ^)3 = a3 + 3 a2J + 3 a J2 + ^3 = 3 C^o^^H 3 C^i^^^ + 3 C>52 + 3 (7353,

(^a + by=a^-\-4:a%-{-Qa^'^-{-4:ah^-{-b^

We shall now show that these developments can be extended
and generalized.

Let n be any value for which we have verified that

(a + by= , 6>" + , O^a^-'b + . . • + n C,a"-''5'- +...+, C.,b\

and let us see what will follow.

Multiply both members of this equation by a-\-b. We
have (first multiplying by a and then by b and adding and

124

BINOMIAL THEOREM 125

remembering that ,1^0 = n+\0(^^ and „C'„ = n+i^«+n since each
is equal to one, and that

n^r "i" n^r-l = n+V-'ri (159, 162})

+ 72^0

«-^-M5'-+...+„(7„6'^+i

or (a-\-by"+^

Therefore (a + ^)„+i has, when expanded, exactly the same
form with respect to 7i + 1 that (a + h~)^ had with respect to
n ; and therefore the development is true for a value of n
one greater. Since we have verified it for n = 4, it is true
by the above proof for n = 5 ; and since it is true for n = 5
it is true for n = 6, and so on without limit. Hence it is
true universally for a positive integral value of n.

Remark. The form of proof here used is known as Mathematical Induc-
tion. It consists of the following steps. By trial in a few cases, which may-
be called the first, second, third, etc., cases, we may suppose that we have
discovered a law. It is next assumed that this law holds for the wth case
in order to build up, on this assumption, the {n + l)th case. If the (n + l)th
case has the same form with respect to w + 1 that the «th had with respect
to w, it can be concluded that if the law holds for the ?ith case, that is, if the
assumption was true, it holds for one more case. But by actual calculation
the law was seen to hold for, say, the third case. Hence it holds for the
fourth, therefore for the fifth, etc., universally.

It must be clearly borne in mind that this mathematical induction is not
the induction of logic and the physical sciences, but that it yields absolute
certainty and universality in the conclusion. For additional work on this
topic see 352-356.

170. It will be shown later that a development similar
to the above holds under certain circumstances when n is

126 COLLEGE ALGEBRA

not a positive integer. In order to state tlie form of this
development we introduce a new notation, viz.,

7i(^ — l)(y^ — 2) •" (n — r + 1) _ AA ^^^

1.2-3 ... r ~\rj ^ ^

for any value of n whatever. The numbers defined by the

symbol [ ) for any given value of n whatsoever, and the

different values of r, viz., 0, 1, 2, •••, are called binomial
coefficients. It will be noticed that if 7^ is a positive integer

and r > n, ( ] = n^r'i i^ ^ is a positive integer and r':>n,

( ^ J = 0 ; if n is not a positive integer, I ] can never become

zero.

171. The development before mentioned, when n is not a
positive integer, will, provided a is numerically greater than
5, be proved in 221 to be

(a + by = (^\i^ + h^ a^-^b + (^^\ a^-%'^ + • • •

+ r^y^-^^*- + ••• to infinity. (3)

This is the general development of (a -f- by\ of which our
previous form (1) is a special case, for when ?i is a positive
integer, (3) becomes identical with (1) and takes the form

(a + by = (^\a- + (tja-n^ + Q)«"-262 + ...

Jr(''\a--'b^^-\-''- + ('''\b^' (4)

The notation just introduced will hereafter be used.

BINOMIAL THEOREM 127

172. Attention is here called to the fact that corollaries
1 and 2 of 157 and the theorem of 162 may be restated and
generalized with the aid of the new notation as follows :

(:)=

n \n — r -}- 1 Ai + l^ f ti \n + '\.

r — IJ r ' \ r J \r —IJ

The proof is identical with that given before, n now being
any real number whatever. Let the student rewrite the
proof with these symbols.

173. In regard to developments (3) and (4), it may be
observed that :

1. In both developments the exponent of a in the first
term is n, and decreases by one in each succeeding term ;
the exponent of h is zero in the first term and increases by
one in each succeeding term; the sum of the exponents of
a and b in any term is always 7i.

2. The coefficient of the first term is unity ; that of the

second is obtained from the first by multiplying it by - ,
that of the third by multiplying the second coefficient by

-— — , and in general that of the (r + l)th by multiplying

ji f I "1

the rib. coefficient by -^^-, since

r

rj \r — 1/ r

3. If in (4) we replace 5 by — 5 Qn being a positive in-
teger), the development becomes

128 COLLEGE ALGEBRA

Ca - by = (^y + Qa^-i(^-b) + g^a-2(_ ^)2 + ...

+ (''V"-^(- hy + ••• + (^\- by

and we see that the sign of a term is + or — according as
b occurs to an even or an odd power in that term.

4. The sign of a term in (3) depends jointly on the sign
of b and the numerical coefficient, and can be determined in
any particular case for given values of b and n.

5. Equations (3) and (4) assume the following form when

(1 + a;)^ = 1 + nx + ^^^ ~^x'^+ ...

2 !

^ n(n- l)(n - 2) ■'• (n - r -hi) ^r ,
r !

174. EXAMPLES

Expand :

1. (a-\-by. 6. (2:2-2 2/3)5.

3\3

2. (a -by. 7. 2;2-

1/

3. (a + 3^)4. 8. (x^ + 2a)4.

4. (a; -2^)6. 9. fa:^ + ^"^^

5. (2 a- 5 5)3. 10. (rt2 + 3?,)i to four terms.

11. Vl — x"^ into a series giving four terms.

BINOMIAL THEOREM 129

THE GENERAL TERM OF (a + 6)"

175. The (r + l)th term in (3) or (4) is ^^^"ja""^^^ or

—5^ ^^-^^-— — — ^ ^^ ^a^ ^o"^ and may be considered

1 . 2 • • r -^

the general term from which each of the other terms may be
obtained by giving different values to r. It may be ob-
served that the number of the term is always one more than

the lower number in the symbol [ J •

176. Example. To find the coefficient of a;~2o in / -) .

/^\ V^ ^/

The (r + l)th term of (a + hy is I ja"-''b\

x^ 2
Here a = — , 5 = -, and n = 15, therefore the (r + l)th

term of g - ^'' is (^^"^ gj "■(- |J. In this expression

^30-2r

we find the factors containing x to be — - — = x^~^^ ; but r is
to have such a value that this exponent shall be — 20.
.•.30-5r=-20, 5r = 50, r=10.

Putting this value of r in the preceding expression for the
(r+ l)th term, we get

15 14. 13 12. 11 .21O0:

-20

1 . 2 . 3 . 4 . 5 . 35

1025024 ^_,o

81

1025024

Therefore the required coefficient of a: ^o is

81

130

COLLEGE ALGEBRA

SOME PROPERTIES OF BINOMIAL COEFFICIENTS
177. Several properties of binomial coefficients have already

been given, as

n

71+1

r

n + 1
r

n
r — 1

71

r — 1

= ( ") +

J J

n — r — 1
r

n + 1

n
r —1

(172)

and if n is a positive integer

n

71 — 7'

(161)

178. From this last statement we derive the following
theorem : Iti the expansioTi of (a + ^)", whsTi n is a positive
i7iteger^ the coefficients of terms equally distant from the begin-
ning and the end are equal ; for^ by the principle just stated.

n
n

n

71 — \J

n

n — r.

179.

Find :

EXAMPLES

1. The 7th term of {x + y)!^.

2. The 5th term of (3 2: + 2 yf,

3. The 4th term of (1 - xy.

4. The 1th term of (1 — x^~'^.

BINOMIAL THEOREM 131

Find the (r + l)tli term of:

5. (1-xy. 7. (1-xyK 9. (1-xy^.

6. (1-xy. 8. (l-a;)-2. 10. (1 - x^.

Find the coefficient of :

11. x^^ in the expansion of (^^ + 2)^^.

12. x^ in the expansion of (a:^ -f- 2 xy^.

13. a; in the expansion of (.-r^ — ^— ) •

14. a:""^ in the expansion of ( ^ ^ ] •

/ 1\12

15. The term independent of a; in i 2 a: ] •

SimjDlify :

16. (3 + Vi)6 + (3 - V^/. 17. (:r+ V3)5+(:r-V3)5.

Find the middle term in :

18. (^ + ^Y'- 19. Cx-{-x-^y\ 20. (^' + ^^'"

21. Find the middle terms of ( 2 a —

aj

22. In the expansion of (^1 -\- x^^ the coefficients of the
(2 r + l)th and (r -f- 5)th terms are equal ; find r.

23. Find n when the coefficients of the 14th and 20th
terms of (1 + xy are equal.

24. Find the relation between r and n in order that the
coefficients of the (r — 6)th and (3 r 4- 2)th terms of (1 + ^0^"
may be equal.

132 COLLEGE ALGEBRA

180. The greatest coefficient in (a + 6)", when ti is a posi
tive integer, is, by 163, \Z) when n is even, and l^\ = (^^

when n is odd, the number of the term being ■^ + h when

n is even, and when n is odd, two terms, the ( — - — Jth and
^"'" J th, having equal coefficients, which are greater than

that of any other term.

181. Theorem. The sum of the binomial coefficients when n
is a positive integer is equal to 2".

Proof. If in the expansion of

we put a and h each equal to one, the right member becomes
merely the sum of the coefficients, since any power oi a oy h
becomes equal to one, while the left member becomes 2" and
expresses the value of the sum ; thus

Corollary. Since f J = 1, the above expression gives

a theorem in combinations, which has been proved in 167 in
a different way.

BINOMIAL THEOREM 133

182. Theorem. When n is a jjositive integer^ the sum of
the hinomial coefficients of the odd terms is equal to the sum of
those of the even terms.

Proof. If we put a = 1, ^ = — 1, in the expansion of
ia + hy = (^^ a- + (^^ a-^h + • • • + (^\%

we obtain ^ (n\ fn\ . fn\ fn\ .

"'■ ^HH> -^HH^- ■■■■

183. Problem. Find the greatest term* in the expansion of

(3 a; + 4 yy when x = b and y = 3.

The (r + l)th term or t,^^ = ("^V3 2^)7-^(4 ?/)%

j^^7-r+l 4^^ (172)

C r 2tx

, _,8-r 12__, 32-4r
r lo or

* By the greatest term is meant the term having the greatest numerical
value independent of its algebraic sign. To find the numerically greatest
term we therefore proceed as if all the signs were positive. Thus the numeri-
cally greatest term in (x — 4i/)i9 when x = 2 and y = 3 is the same as the
numerically greatest term in (x + 4 yy^ for the same values of x and y.

134 COLLEGE ALGEBRA

32 — 4r
and ^^+1 > ^,., while r increases so long as > 1, and no

5r

longer. That is, so long as

32 — 4 r > 5 r,

32 > 9 r,

r<3f.

Therefore, since r must be a positive integer, the greatest
value which it can have which satisfies the inequality 32 —
4r>5r is 3. Therefore the greatest term is the 4th.

EXPANSION OF A MULTINOMIAL

184. The expansion of a multinomial may be obtained by
grouping the terms of the multinomial into the form of a
binomial. Thus :

= (2;2- 3 0^)3+3(2:2-3 ^)24 + 3(2:3_ 3^)42.^43

+ 482^-144a:

+64

^a;6_9^5_^39^4_ 99 a;^ + 156 x^- 144 x + 64.

To obtain the general term^ or term containing a-^m^m^^ . . .
aj'-, in the expansion of (a'j + ^2 + ^3+ ••• + ^,.)", we observe
that (^a^ + a^-\- -•■ +«,.)'*= (^^ + ^2+ ••• +«,,)(aj + a2+ ••• +a^r)
••• (^a^ + ^2 + ••• + a,.') to n factors and is homogeneous of de-
gree n. To form the given term we may pick out a^ from

Zj of the n factors in ( J ways, and after that we may pick

/ _ 7 \
out ^2 from ?2 of the remaining n— l^ factors in ( 1 ] ways,

BINOMIAL THEOREM 135

and by associating all the ways of making the first selection
with all the ways of making the second, we may select a^, l^

times and a^, I^ times together in f V M ways; that is,

we may select a^^^a^^- in r j( M ways. Similarly from

the remaining n—l-^ — l^^ factors we may select a^ from l^

factors in y~ ^~ 2j ways; proceeding in this way we see

that the number of ways in which the product a-lm^m^^ ••• a^^^
can be formed is

(n — I^ — I^—-'- — Ir-iVi _ nl

n

where l^-\-l^-\- ••• + ?,. = n. The number ' — — is the

number of ways in which the product a^m^m^m^* "- a,!r
appears in the expansion (^a^-\-a^-\-a^-{- •" -\- a^y\, therefore
it is the coefficient of that term, and lience the general term
in the expansion of (a^ + a^-[- a^-^ ••• + <^;)" is

•a/i«o2 ••• a Jr.

To find the coefficient of x'' in (^q + a^x + <^2^^ + ••• + «r^0"*
The general term by the preceding section is

n\

ft '1*^' ••• I/,-, ,

aQo(a-^xy^(a.-^x^y-. ••• (a,^'')'»-

136 COLLEGE ALGEBRA

Therefore to find the coefficient of x^ we must find all inte-
gral solutions satisfying both the equations

^1 + 2/2 + 3/34- ••• -Vrl^.^h.

To every solution corresponds a term in 2;^. The col-
lected coefficient of x^ is the sum of all the partial coefficients
corresponding to the several solutions of both equations.

Example. Find the coefficient of a;^ in (\-{-x-^x^-^ x^~)^.
We have to solve the equations

'0 ^~ ^1 "'" 2 ~l~ ^3 ^^ ^'
?i + 2 Z2 + 3 ?3 = 6,

and find as solutions for Z^, ?j, l^^ Z3, and the corresponding
partial coefficients,

^0'

h

^2'

h-

Partial coefficients.

0

1,

1,

1

3! _g

^1

0!1!1!1!

0

0,

3,

0

- ^- -1

^•>

0!0!8!0!

1,

0,

0,

2

3! _

-f t rx 1 rx 1 ^1 ^1

the sum of which is 10. Hence 10 is the coefficient of x,

EXTRACTION OF A ROOT

185. The following example shows how the root of a num-
ber may often be extracted with advantage by means of the
expansion (3), 171.

BINOMIAL THEOREM 137

-^/63 = 63^ = (64 - 1)' = 64* - 164-<^ + iilnl) 64-^^..
^ ^ 6 2!

6 ' 25 72 * 2"*"
= 2- 0.005208 - 0.0000839 ...
= 1,9947577+ ...
= 1.994758.

186. EXAMPLES

Find the sum of the coefficients of:

1. (x + ^y^. 3. (x-\-2i/y. 5. (x — 2'i/y.

2. (x-^y\ 4. (x-2ijy. 6. (2a; + 3^-y.

7. A man invites 9 friends to dinner 1 at a time, 2 at a
time, and so on up to 9 at a time. How many different
parties does he form?

Find the numerically greatest term in :

8. (^x + I/)", when x = 2 and ?/ = 3 ; when x = 2 and

9. (2; — ^)^S when x =S and ?/ = 4.

/ 1\15

10. ix^+-j , when x = ^ and «/ = 3.

11. (a: — 3 ?/)i^, when x=l and ?/ = f .

12. (2^4- 5)20, when a = |.

13. (2 a + 3 by\ when a = 2 and ?> = 5.

138 COLLEGE ALGEBRA

Expand :

14. (x-^y + zf. 15. {x^ +2x-^f,

16. Find the coefficient of x'^ in (1 -\- x -\- x'^ + a:^)^.

17. Find the coefficient of a;^ in (1 -{- ^ x + x^ -\- 2> x^Y.

18. Find the coefficient of x^'^ in (2 —x + x'^ -\- x^y.

Find to 5 places of decimals :

19. V^. 20. a/80. 21. a/125.

CHAPTER X
CONSTANTS, VARIABLES, LIMITS

187. Definitions. A constant * is a number which during a
given discussion does 7iot change in value.

A variable is a number which under the conditions of the
problem may assume various values during the same discussion.

Thus the circumference of a given circle is a constant,
while the perimeter of a regular polygon of 7i sides inscribed
in it is a variable dependent upon n.

188. Definition. If under the law of its change a variable
number can be made to approach as 7iear as we please to some
constant number tvithout becoming ec^ual to it^ the constatit is
called the limit of the variable.

Let us use the symbol ^ to signify " approaches as its
limit," and Lx = a to signify the limit of x is a. Thus x = a
is read "a; approaches a as its limit."

In the previous illustration the circumference of the circle
is the limit of the perimeter of the regular polygon, as the
number of sides becomes indefinitely great; the area of the
circle is the limit of the area of the polygon.

189. The sum of the geometric series 1 + - + -H is a

variable whose limit is 2. This can be proved by con-
sidering the formula for the sum of n terms of a geometrical

* Constants are sometimes absolute, such as 2, 3, •••, and sometimes
general, as r, 7i, •••.

139

140 COLLEGE ALGEBRA

(tCA T^^ CI CLT^

progression, s = — ^- ^=i- . As n becomes

indefinitely great, s is a variable and approaches as its

n 1 — r

CIV

limit, since approaches zero when r is less than 1.

1 — r

Thus 1 + - + - 4- • • • approaches = 2 as its limit.

2 4 1—2

The same can be shown geometrically as follows : Take a
line AB

A I ! ' H— *— »B

Fig. 26.

2 inches in length. Take half of it, then half of the re-
mainder, and continue the process, always taking half of
each succeeding remainder, and the sum of all the parts so
taken will represent the given series. This sum evidently
approaches as near to 2 inches as we please, but under the
law of formation cannot reach it.

190. We state here for convenience the following familiar
theorems without proof. A treatment of limits together
with the proof of these theorems is found in 357-370.

1. If two variables are constantly equals and each approaches
a limits their limits are equal.

2. The limit of the algebraic sum, of a finite number of vari-
ables is equal to the algebraic sum of their limits.

3. The limit of the product of a finite number of variables
is equal to the product of their limits. The limit of a constant
times a variable is the constant times the limit of the variable.

4. The limit of the quotient of two variables is equal to the
quotient of their limits.

191. In the following examples, which illustrate the use
of limits, it is to be understood that {f(x)}^^^ = L\f(x)}.

Ex. 1. Find L 2^'-g^ + 2

x=f) OX^ -{-i X— D

CONSTANTS, VARIABLES, LIMITS 141

By 4, 190, we have

o 2 Q , o L (22;2-3a:H-2)

a; = 0

= -?. (by 2, 190)

o

Ex. 2. Find L ^^-^^ + ^.
x=oo02:^ + 7 a; — 5

Dividing both numerator and denominator of the fraction

by x^. we have o o

2-'^ + 4
^ 2^_Z^^+_2^ ^ X x^

a;=Qo 5a;2 4- 7 a; — 5 a-^oo r , 7 5

0-1 ^

a: X''

= -. (by 4 and 2, 190)

5

^^ g ^ (2 2;+5)(3rg-7)(2-3a;)
:r^o (rc2+5)(2:z:-7)

i(2a;+5)-i: (3a;-7)-X (2-3:r)

= £±0 x=o x=o (by 4 and 3, 190)

L (2;2+5).Z (2a;- 7)

a; = 0 x = ^

_ O^X-'^X^) = 2. (by 2, 190)

5(-7)

, J (2a: + 5)(3a:-7)(2-3a:)

.t^ (a;2+5)(2a:-7)

r2 4-^V3-IY?-3

J \ Xj \ X J \X /^ Q

Q?'j\ Xj

142 COLLEGE ALGEBRA

by dividing numerator and denominator by a?^ distributed
among the different factors according to their degrees, and
by taking limits according to 4, 3, and 2, 190, as in Ex-
ample 3.

■r^„ c J (a: + o) ^ j^ V ^^

V xj\ X x^Jx

= oo

(1) (1) (0)

by dividing numerator and denominator by a;*, the highest
power of X found in either, and completing as in Example 4.

Ex. 6. Find the value of ■ when x = a.

X — a

If we put X equal to a in this fraction, it assumes the form

of -, which is called an indeterminate form, because in this
0

form its value is not apparent. Its appearance does not
show that it has no definite value, but that we have not
used the proper method to obtain that value. So long as

x^a^ = 1, therefore its value does not depend on x^ and

X— a

hence it is unity for all values of x.

Ex. 7. Find the value of (tpAl±3\ .

If we put a: = 2, the fraction assumes the form of - .

We give three solutions of this problem.

(1) We may factor numerator and denominator and obtain

a;2_5^-|-6_(a;-2)(a:-3)_a;-3

.2

2 — 3 1

which for a; = 2 is equal to = -

^ 2-4 2

^-__6a;+8 (a;-2)(2;-4) x-\'

CONSTANTS, VARIABLES, LIMITS 143

^"■^ ^2- 6 re + 8^=2 ^=^2(a:-2)(2;-4) 2 2'

by 3, 190.

(3) Another solution is obtained by putting x= 2-^ h and
taking the limit when ^ = 0 ; since as a; = 2, A = 0. Then

^Y2:2-52: + 6\ ^(2 + /02-5(2 + 70H-6^ ^ Ji^ -
z^2W - 6 a: + 8y A^o(2 + hy - 6(2 -f- A) + 8 h^oh^-2

h

h^^h' h-2 2*

In the preceding example we found the expression - repre-

1 ^
sented the value 1, in this example it represents -, and it is

clear that in general its value depends upon the manner in
which it has arisen.

Ex. 8. Find the value of ^ V-^— v^ + V^ - 2 a

If we put x=2a^ this assumes the form of - . We shall

use two methods of avoiding the difficulty. First, by factor-
ing. We notice that we can separate the fraction into two
parts :

VJ— "\/2~a 1 Va:— 2 a
and .

Vrr^ — 4 a^ Va;2 — 4 a^

Since Va:^— 4 ci^ = ^{x + 2 a){x — 2 a)

and Va;— 2 a =-^(V2:4- V2 a)(Va; — V2 rt),

144

COLLEGE ALGEBRA

we have

V(V2:-V2a)2

'^x — V2 a\ _ __
Va;2 - 4 a?)x=2a \y\(x+2a') ( Vi4- V2a) (Vx- V2^)^=«=2«

VVa;-V2

a

0

=2a

and

-\Jx— 2 a

V4a(2V2a)

Va; — 2 «

0,

V2;2 - 4 a2y^2a V V(aJ + 2 a) (2J - 2 «)^^=2a V4 a 2 Va '

-\fx — '\/ 2 a -\- '\/ X — 2 a

hence

Va;2 — 4 a^

^2a 2Va

Second, rewriting the original fraction with fractional ex-
ponents, and putting a; = 2 a + A, and noticing that when

a: = 2 «, 7i = 0, we have

■\/x--^I^^-\/x-2a\ ^/(2a+A)^-(2a)i + Ai

/^2a V (4«A + A2-)i

V,

x'-

4«2

/i=0

and expanding the first term of the numerator by the bino-
mial theorem, and factoring the denominator, we have

(2a)i+l(2a)-U4- (2a)^ + A^

7i2(4a + A)2

^=0

l(2a)-UH- ••• +A^

A2(4a + /0^ j^=o

CONSTANTS, VARIABLES, LIMITS

145

Dividing both numerator and denominator by A^, we have

(2a)-^Ai+ .•• +1

(4 a + hy
Ex. 9. Find the value of

^ /i=o (4a)2 2V

a

(

V«^ + aa: + a;2 — Va^ — aa; + a;^

d:=0

Multiplying both the numerator and the denominator by the
conjugate surd of each, we have

2 ax{;\/ a -\-x-\- ^ a — x)

2 2:( Va^ -}- ax -\- x^ -\- Va^ — ax-\- x^}

as the value of the fraction for any value of x, and its value

2 a

i^" — v'

192. To find the value of

\ U— V Jv=u

We distinguish three cases :

1. When n is a positive integer.

2. When n is 2i positive fraction.

3. When n is negative whether integral or fractional.

1.

W'' — V
U — V Jv=u

= (i^"-lH-W"-2v + W«-3|;2+ ... _|-t,«-l)^,^^

= nu^ ^

2. When n=^. Put u'^ = x, v*^ = y, then
9

p p
u — v

x*-

r\ _

X""

1 L.-,

yvy=.

X^

y

p^

r'l — )i'l

_px

•p-1

qpC

9-1

146

COLLEGE ALGEBRA

by case 1. Multiplying both numerator and denominator by
x^ we have

p p

U — V

q x'^ q u q

3. Put n = — m, then

71

-m ny—m

^m _ ^m

II — V Jv=u \U'"V^'\U — Vyjv=u

VIM

m—l

u-
= — mu

— m—l

We see that in each case we obtain a result of the same
form in terms of n.

193. EXAMPLES

Find the limits when a; = 0, and when x = cc^ of :

\ — x^ ^ 1 — X

2a^-l 2

x^

(^-x)(x-h^)(2-lx)
(lx-l)(x + iy

Find the limits when n = 0, and when n= cc, of :

3.

4.

n

n — 1
n + 1

n

5.

6.

a + (ri -

-l^d

a + (^n -

-2)d
1

^2 (n_l)2 (2n + l)(27i + 2) (27i-l){27iy

x^-{-l

7. Find L

x^-l X

2-1

8.

Find the value of :

\a^ — .r^)^ ^(^a — xy
{a^ — x^y + (a — a:)2^^=«

a^ 4- a;^ + a:^ + 1

2;

-1

x=0

CHAPTER XI
SERIES

194. Definitions. A series z'.s a succession of terms each of
which is formed according to the same law. If after a certain
number of terms the series comes to an end, it is called ^finite
series. If the terms continue in an endless succession, it is
called an infinite series. The binomial series when ^ is a
positive integer is an example of a finite series. When n is
not a positive integer, the series becomes an infinite series.

If a series is finite, the sum of its terms is manifestly some
finite number. But if a series is infinite, the sum of the first
n terms as n increases is a variable dependent on the value of
n^ and in the case of different series will, in general, approach
different limits.

195. Definitions. A convergent series is a series in which
the sum of the first n terms approaches a finite and determinate
limit however great the value of n may be. We define the sum
of a convergent series as the limit which the sum of the first n
terms approaches when n becomes indefinitely great. Thus the
geometric series l + J + i + J+ ••• approaches the limit 2, is
therefore convergent, and 2 is the sum of the series (189).

A divergent series is a series in ivhich the sum of the first n
terms does not approach a finite and determinate limit as n he-
comes indefinitely great.

If a convergent series, some of ivhose terms are negative., re-
mains convergeiit wheji all its terms are made positive^ the series
is said to he absolutely convergent.

U7

148 COLLEGE ALGEBRA

196. Theorem. Beginning with the nth term of a convergent
series^ the sum of any number of consecutive terms approaches
zero as a limit as n becomes indefinitely great.

Proof. Let the series be denoted by

the sum of n terms by aS'^, and the remainder after n terms
by En- Then

Hence ^n+m *^n-l = ^» + ^/i+l + ••• + '^n+m*

By hypothesis L tS^ = S; (195)

n=oo

therefore L (^S^^^^ — S^-i) = S — S=0,

and therefore L (^^„-f Un+i + ••• + w„+„J = 0,

for any value of m.

Corollary 1. In particular when m = 0 we have

L (i^O = 0.

n=oo

That is, the nth term of a convergent series approaches zero as
a limit when n becomes indefinitely great. It is to be carefully
noted that this is a necessary but not a sufficient condition for
a convergent series.

Corollary 2. If the nth term of a series does not approach
zero as a limit whe^i n becomes indefinitely greats the series
cannot be convergent and is therefore divergent.

SERIES 149

Corollary 3. The remainder 'after any numher of ter-ins
approaches zero as that numher of terms becomes indefinitely
great. For when m = go in the expression

■^ y^n ~r ^n-\-\ + *•• H~ ^«+my»
n=oo

we have the remainder after n—1 terms, and we have shown
that this expression has zero as its limit.

Corollary 4. Conversely, if the remainder Rn after n terms
approaches zero- as a limit as 7i becomes indefinitely greats the
series is convergent. Since by taking n sufficiently large the
possible further increase or decrease of Sn can be made less
than any assigned quantity however small, therefore S^
approaches a finite and determinate limit, and by definition,
195, the series is convergent.

From Corollaries 3 and 4 we observe that the condition
that R^ approaches zero as its limit is both a necessary and
a sufficient condition for a convergent series.

197. If the series which begins with a given term of a
given series is convergent, the entire series is convergent by
the definition of a convergent series, and conversely, 195.

And if the series which begins with any term of a given
series is divergent, the entire series is divergent by the
definition of a divergent series, and conversely, 195.

198. If a series consisting of positive terms is convergent,
any series consisting of positive terms which are as small as
the corresponding terms of the first series, or a series formed
from either by taking any or all of its terms with negative
signs, is convergent. For in either case the remainder after
n terms will be at least as small as or numerically smaller
than that of the first series, hence this remainder will also
approach zero as a limit, and tli,eref9re either. series will be
convergent. T" \ "T^ V

_-. _ - -\ -■■

150 COLLEGE ALGEBRA

199. As explained in 120, when x is real, the symbol \x\
denotes the numerical or absolute value of x ; thus, | — 2 1 = 2
and I 2 1 = 2.

200. We shall now consider the convergence of the geo-
metric series -too -,

l-i-x-\-x^-{-x^+ '•' -i-x^'-^-h ....

By division, or by the theory of the sum of a geometric
series,

l-{-x-\-x^-\-a^-\- ••• 4- x^^~^ =

l-x"" 1 x"

- X 1 — X 1 — X

There are three cases to consider according as

|2:|<1, |a;|>l, or I a; 1 = 1.

First, when |a^| < 1. In this case x^ = 0 as n becomes in-
definitely great. The denominator, 1 — a;, is constant. There-

fore = 0. Hence the limit of the sum of n terms as n

1 — X -|

becomes indefinitely great is , which is a finite and

1 — X

determinate limit. Therefore by the definition of a con-
vergent series, 195, the series is convergent.

Second, when |a7|>l. In this case x"^ becomes infinite as
n becomes indefinitely great ; and therefore the sum of 7i
terms does not approach a finite limit, and therefore the
series is divergent, by 195.

Third, when x= -\-l. In this case each term is unity and
the sum of n terms becomes infinite as n becomes infinite,
and the series is therefore divergent.

When x= — 1 the sum of 7i terms is alternately 1 and 0,
and therefore the sum of n terms does not approach a deter-
minate limit. Therefore the series is by definition divergent.

This last kind of divergent series is called an oscillating
series.

SERIES 151

201. Definition. An oscillating series is a divergent series
in which the sum of n terms, though always finite^ does not
approach a deternmiate limit.

202. The question of the convergence or divergence of a
given series is of the utmost importance. For in the subject
of mathematical physics and in other branches of applied
mathematics it is usually necessary to throw a function into
the form of a series in order to calculate its value. In order
that the series may correctly represent the function, it is
necessary that it be convergent. The danger of using a
divergent series to represent the function is evident by

throwing the function into the form of a series by actual

division. We obtain

= 1 + 2- + 2;2 + a.^ + ... -f .T« + ....

1-x

We have just shown that when |.r|<l this series is con-
vergent and equal to the first member. When | a: | > 1 we
have shown that it is divergent, and if w^e should attempt to
use a; = 3 in the above equation we should have the absurdity

-1 = 1 + 3 + 9+ ... =00. •

Hence the physicist first examines his series with respect
to convergence or divergence to see if it is safe to use it.

METHODS EOR TESTING THE CONYERGENCY OR DIVERGENCY

OF A SERIES

203. First Method. If in a series the numerical value of
each term is greater than the same number e hoivever small, the
series is divergent.

The proof of this theorem may be seen by observing that
it is a special case of failure to satisfy the conditions of

152 COLLEGE ALGEBRA

Corollary 1 of 196, and therefore is a special case of Corol-
lary 2 of 196. For by the latter corollary any series what-
ever, in which the nth term does not approach zero as a
limit as n becomes indefinitely great must be divergent.

Example. Test the following series for convergence
or divergence :

5+^ + 5+....
5 7 9

Here we observe that the several numerators are in an
arithmetical progression of which the nth term is n-{- 2;
likewise the several denominators are in an arithmetical
progression of which the 7ith term is 2 ?^ + 3. Therefore the

n-{- 2

nth term of the series, u^ =

2ri + 3'

But .^^>;^ + ^

2^+3 2^ + 4'

that is, — ltj:L > _

2n-\-S 2'

for all values. of n; therefore the series is divergent.

204. Second Method. A series of alternately positive and
negative terms m which each term is 7iumerically smaller than
the preceding^ and in which the nth term approaches zero as a
limits is convergent.

Proof. Let the series be

S = u^ — n^ -\-u^ — ?/^ + • • • ,

then we have

S =(u^ — u^ + (u^ — u^ + (^^5 — We) H , (1)

and also S = u^ — (u^ — ?^3) — (u,^ — Wg) — •••, (2)

SERIES 153

where each parenthesis, in itself, represents a positive number,
since by hypothesis each term is numerically less than the
preceding.

Form (1) represents jS as the sum of positive numbers,
and shows that it is greater than Wj — ii^. Form (2) shows
that S is less than Wj. Therefore S is positive and finite,
and lies between Wj and u^ — u^.

Again, since

we have | i^„ | = Un+i — u,^+^ + w^+g — u^^^ + • • • ,

whence |7^,J can be thrown into the two forms corresponding
to those above for S.

Thus, \Bn\= (w„+i - Un+2) + (^„4-3 - %i+0 + • • • , (3)

and also \Rn\= ^n+i - (i^n+2 - '^^+3) " (^^«+4 " w«+o) ' • '• (4)

Therefore |i2„| lies between \un+i\ and |i*„+^ — w^+gl* ^^^
\Un+^\ and \Un+i — Un+2\ botli approach zero as a limit when n
becomes indefinitely great. Therefore \Rn\ approaches zero
as a limit when n becomes indefinitely great. Therefore,
by 196, Cor. 4, the limit is determinate and the series is
convergent.

Example. 1 h- — - + - f- ••••

2 3 4 5 fc)

Here the nth term, w,^ = (— 1)"~^-. Therefore the nth

n

term approaches zero as its limit as n becomes indefinitely
great, each term is numerically smaller than the preceding,
and the terms are alternately positive and negative. Hence
the series is convergent.

154 COLLEGE ALGEBRA

205. Third Method, or Method of Ratios. This method will
be considered, under two theorems:

I. Theorem. If m an infinite series heginning with and
after a certain term the ratio of each term to the preceding is
numerically less than a positive number which is itself less than
1, the series is convergent.

Proof. First, when all the terms of the series are positive.
Let the series beginning with this certain term be repre-
sented by

>S = Wj + 2^2 + ^^3 + ••• + ^r + ••••

But '^<k,'^<k,'^ <Tc, ...,'^ <h, ^^<k, •••,-?;<l.

U-^ U^ Uq '^/i-2 '^^n-l

Therefore, by multiplying the first two of these inequalities,
then the first three, and so on, we have

% ^2 ^ 1,2 ^4 ^3 ^2 ^ L3 '^^ ^^-1 ^4 ^3 '^h ^ Z,n-1

Un U-l Un U(f U-t U^i_-l Uf^_s) 11 o Un U-i

or, '^<k\ '^<k\ ..., ^<F-i, .-.

Therefore, by clearing of fractions,

U-\ — U-i,

Un "^ U-irCtf
Wg < U^'^^
U^ < U-J^^

u'' < u^V-\

SERIES 155

that is, the series S is less, term by term (except the first),
than the series

jS' = u^ + 2i^k + u-Jc'^ -\- ii^lc^ -\- ... -\- u^k'^-'^ + •••
= ?/i(l + ^ + A;2 + F+... +V-^+ ...)•

But the series 1 -{- k +k'^ -{- k^ + k''~'^ + ••- approaches a
finite and determinate limit, by 200 ; therefore S' = Lim SJ

= Lim 2^^ (1 + y^ 4- A;2 + F + . . . + A;"-i) = u^ (t^-t) . by 190, 3,

is finite and determinate. Hence S' is convergent, 195, and
JIJ = 0, 196, Cor. 3 ; but since S is less, term by term, than
aS^', Rn<Rn^ and so much the more B,,^ = 0 as n becomes
indefinitely great, and S is convergent, by 196, Cor. 4.

Second., if the terms of S are not all positive, the series
will still be convergent, by 198. And in each case it follows,
by 197, tliat the entire series is convergent.

II. Theorem. If in an infinite series^ beginning with and
after a certain term^ the ratio of each term to the j^recediyig is
numerically equal to or greater than 1, the series is divergent ;
for after this certain term the terms are either equal or in-
creasing numerically, therefore the nth. term cannot approach
zero as a limit Avhen n becomes indefinitely great, and by 196,
Cor. 2, the series is divergent.

206. The method of 205 is applied in practice as follows :

If the limit of

u.

^n-\

as n become indefinitely great is less

than 1, the series is convergent, for the hypothesis of Th. I
is satisfied. If this limit is greater than 1, the series is
divergent, for the hypothesis of Th. II is satisfied. If the

ratio — — is greater than 1 and approaching 1 as its limit,

the series is also divergent, by Th. II. But if the ratio

■«-i

156 COLLEGE ALGEBRA

is less than 1 and approaching 1 as its limit, the hypothesis
of Th. I is not satisfied, and therefore we cannot decide by
this method whether the series is convergent or divergent.
In this case the ratio method is said to fail, and we require
another test for such series.

17 . 3,3-4, 3-4.5 , , 3-4.--(n+2) ,

Example. - + _- + -— - + --- + -— — /:, .-,{ + •••.
4 4-b 4-6-8 4-b---(2^i + 2)

Here u . _ 3 • 4 ■ 5 - (» + !)(» + 2)

Here **»- 4 . 6 . 8 - (2»)(2«+ 2) '

from which Un_^ can be obtained by changing n into n—1;

thus, Un-i

3.4.5... (n+1)
4-6 -8..- (2 n)

Hence, by dividing and simplifying,

u„ _ n-\- 2
u^-i 2 /i + 2

Taking the limits, X -^ = ^ • (191)

Therefore the series is convergent.

207. Fourth Method, or Method of Comparison of Two
Series.

Theorem. If Si^=u^-{- u^-\- -" +it„+ ••• and S^, = v-^^-{- v^
+ ••• + ^M + •••? ^^^ ^^^ series coyisisting of positive terms, and if
the ratio — * is fiiiite and 7iot zero, for all values of n whatever,

both series are convergent together, or else divergent together.
Proof. By 39, B> \ ^ ; , — ->r.

SERIES 157

where R and r are the greatest and the least values of the
ratios -\ ^^ ••., !^, and therefore ^h + ^2 + "• + ^" ^ ^

where x is finite and not zero, as n becomes indefinitely
great. Whence u^-\- u^-\- • • • -\-iin = x(y^ + ^2 + " " + ^n)^ »^i^d
by taking the limits of both members of this equation we
obtain 8^ = X - iS„, where JC is finite and not zero, from which
it appears that both series are convergent together, or else
divergent together.

208. The Auxiliary Series. For purposes of comparison in
the application of the preceding method the following series
is of great advantage :

Considering its convergency, it will be found that it is con-
vergent if r>l, but if r>l, it is divergent.

Proof. There are three cases according as r is greater

than, equal to, or less than 1.

1. In the given series the sum of the second and third

2
terms is less than — , the sum of the next four terms is less than

4 8

— , the sum of the next eight terms is less than — , and by

^r yr

continuing in this manner we see that the sum of the given

series is less than ^ . ^

l+lj-l-i-A-i-...
2^ 4^ 8'' '

11 1

that is, less than 1 -\ -\ — - -\ + • • • .

2''"^ ('2^~^)^ (2'~0
But the latter is a geometric series whose ratio is —— ^ and
when r>l this ratio is less than 1 and the geometric series

158 COLLEGE ALGEBRA

is convergent, by 200. Therefore so much the more is the
given series convergent when r>l (198).

2. When r = 1, the series becomes the harmonic series^

1 + 1 + U... + 1 + ....

A 6 li

In this series the sum of the third and fourth terms is greater
than -, that of the next four terms is greater than -, likewise
of the next eight, the next sixteen, and so on. Thus the sum
of the series is greater than 1 + -H !--+••• to infinity,

A A li

therefore the series is divergent (195).

3. When r<l, each term of the series — 4- — + — H

!'• 2'- 3'- 4^

+ ••• is greater than the corresponding term of the har-
monic series, and therefore the series is divergent. This also
includes the case when r is negative.

209. In using the method of comparison it will often be
found convenient to take the series v^ + ^2 + ^'3 + '** ^s the
auxiliary series — -\ 1 — ■4---.. To determine the value

of r that is to be chosen in any particular instance, when the
terms of the series %i^ + n^ + W3 + • • • are algebraic fractions
in terms of w, we introduce the following definition and rule.

210. Definition. Tlie total degree of an algebraic fraction
expressed in terms of n is the degree of its numerator minus
that of its denominator in n. Thus the total degree of

~ — - — is 2 — 3, or — 1. The total degree of — ^ is

n^^b Vn4 + 3

The total degree of the auxiliary series chosen should he the
same as that of the series to he tested. Also in applying the

SERIES 159

method of comparison, we consider L —• If this limit is

71=00 n

finite and not zero, the series are both convergent together,
or else both divergent together.

Example. Ll? +^a;+— ^i;^^ ••• _^^0^+ ^):^»-i + ..
5 12 31 713 + 4

X being considered as positive.

Here u„ = ^ ^x^ S w„_i = — ^^ f- — -x^ ^.

71^ + 4: "^ (^_ 1)34. 4

Hence — n- — l^ 1 — — ja — \ — /^^

Un-l (^3+ 4)0^-1)

and i -^^ = X, (191)

Therefore if x<l, the series is convergent, (206)

and if x>l, it is divergent. (206)

But if 2;= 1, it can be shown that — — < 1 when n>2, and

therefore the ratio test fails. Let us then use the method
of comparison for this case. The total degree of w„ is — 1,
and hence we choose for auxiliary series

z 6 4: n

whose nth. term, v.J^ = -, is of total degree — 1.

n

1 he ratio — ^ = — ^-- — ' — - •

Vj^ ?r + 4

Hence X -^ = 1, which is finite and not zero. Therefore
both series are convergent together, or else divergent together.

160 - COLLEGE ALGEBRA

But the series 1 + ^ + o "I — ^^ divergent. (208)

Hence the series — ^ + ^^ + -— — + • • • + ^'^^ — r^ + • • •,
5 12 31 n^-\-\

or the given series when a; = 1, is also divergent. (207)

211. In testing the convergence of any 'particular series^ the
student should carefully examine the various methods to see
which applies to the series under consideration. When more
than one method applies^ the simplest should he chosen.

212. EXAMPLES

Examine the convergence of the following series :

1 2 3 4 5
2+4+6+8+ •

2 1 + 3 5 1 17 ... . 2''-'+l

+ 4 + 8 + 16 + 32+ + 2« + ■

3. 1_1 + 1_1 + 1_....

3 6 7 9

4. -i "—+ 1

a + 5 a-\-2h a + 36

33-5 3.5-7 3-5. T--. (2^ + 1)

4 4.7 4.7.10 4- 7.10... (3 ^i + 1)

6. ZxA \- h ••••

4 9 16 25

7. a-\-(^a-^d)x+ (^a-\-2d)x^+(ia + ^d)x^-\- •'-.

8. — L_ + _±_ + ^^-|- ... -j ^"^ 1-....

3. 45. 67. 8 (2^ + l)(2*i + 2)

SERIES

9.

i+ 1 + 1 + 1 -

X x-\-\ x-\-2 x+S

f....

10.

I + - + 2V3V-.

12.

1

2! ' 4! *"•

11.
14.

X^ , X"

* 3! + 5! ••••
2 + 8 4 5 ^.._

13.

X-

^ 7? a^
2 3 4

161

4^8 16^32 ^^ ^ 2» ^ •

16. Prove that the series contained in Examples 11 and
12 are absolutely convergent.

17. Prove that the series

-, , , x(x — l^ o , x(x—l)(x—2^ q ,
1 4- ^j/ + 2! ^ ^ 3! ^ "^ "*

is absolutely convergent when | y | < 1.

213. From the definition of 195 the following propositions
result:

1. An absolute! 1/ convergent settles is convergent irrespective
of the order in which the terms are taken.

2. The sum or difference of two absolutely convergent series
is an absolutely convergent series^ and the two series may there-
fore be added or subtracted in any order,

214. Definition. The series formed by taking the moduli of
the terms of a series is called the modular series.

If the modular series of complex terms is convergent, the
series itself is also convergent. This is seen by using the

M

162 ■ COLLEGE ALGEBRA

geometric representation for the addition of complex num-
bers (130), in connection with the principle that the modulus
of the sum of two complex numbers is not greater than the
sum of their moduli, since the sum of two sides of a triangle
is not less than the third. Thus if the sum of the moduli
of the terms of a series is finite and determinate, so much
the more is the modulus of the sum of the series, that is, the

point P of the ratio --— , tlie sum of the series, is a determi-

nate point lying in the finite region of the plane.

Theorem. If the modular series of real terms is convergent^
the complex series itself is, hy theforegoiyig, convergent^ and is
said to he absolutely convergent.

215. EXAMPLES

Test the following series for convergency or divergency:
14 2n

1.

2' 7' '3n + l'

11

7' 4' ' 2n2+a?i + 2'

oil n^

8' 6' ' »3+2w2+3,j + 2'

4 1 1 ... -3

2 '^2 + 2

^ 1 22 71^

5- t:-,

^ 5 + 2^ ^2 + w2 -h 1

SEKIES 163

12 n

O " 1 " ' 1

2 11-2 (7i^+sy

n—\

t

^ V2 — 1 V« — 1 „_-,

VU V/t^-2

Test the series whose nth. term is:

8.

V?i — Vn — 1

^C^ — 1) n-1

10. ^^ "^ X^ 1

(n + V){7i - 2) (n + 8) (n - 4)

CHAPTER XII
UNDETERMINED COEFFICIENTS

GENERAL THEORY

216. Theorem. If the infinite series a^ + a^x -\- a^ + a^:i^
4- ••• -\-a^x^ -\- •••, Inhere the as are constants^ is convergent for
a value of x other than zero^ and is equal to zero for all values
of X which make the series convergent^ then the coefficient of
every power of x is zero.

Proof. If the series is convergent for any particular
value of X not zero, it is also convergent for a value of x
numerically smaller, 198, since each term is thus made
numerically smaller. Therefore it is convergent when x
approaches zero as a limit. Hence we may take the limits
of both members of the equation a^-{- a-^x -^ a^x"^ -\- a^oi^ -{- •••
+ a^x^ -\- ••• =0, when x=0. In order to take these limits
we write the equation in the form

a^ + xS = 0,
where S= a-^-\- a^x + a^x^ + • • •?

and we observe that x - S \^ convergent for all values of x

which make the original series convergent, by 197. Hence

X • S is finite and determinate, by 195 ; if a; = 0, aS' is equal

xS
to flfj, and is finite and determinate; and since S= — , for

X

any permitted value of x^ S is always finite and determinate
for all such values of x^ and therefore aS' is convergent, by
195. Hence x • S approaches zero as its limit when x ap-

164

UNDETERMINED COEFFICIENTS 165

preaches zero. Hence La^-^- Lx - S=()^hy 190, 2 ; therefore

a^ = 0, and since a^ + x - S =0^ x • S =0, and hence as a;^ 0,
aS'=0; i.e. a-^^ + a^x-^-a^x^ -{-'■■ =0. By a repetition of
the same process it follows that <2j = 0, then a^ = 0, and in
general a^ = 0, for every positive integral value of n.

217. Theorem. If two infinite series a^ -\- a-^x + a^^s^ + -- -
-{■ajX^ -\- '•' and bQ-{- b^x + b^x" + ••• +5,^^+ ••• are equal to
each other for all values of x ivhich make both series convergent^
then the coefficieiits of the corresponding powers of x are equal.

Proof. We have given a^ -\- a^x -{- a<f)^ -\- ••• -{- a^x^ -\- •••
= 5q + b-^x + b^x^ + • • • + b^x^ 4- • • • for the specified values of x.

Hence

S— ^0+ («i — ^i)2; + («2— ^2)^+ ••• +(^n— ^n)2:"+ ••• =0.

Therefore, by the preceding theorem,

S~^o = ^' ^j — ^1 = 0, a^ — b^=0, ..., a„ — b^ = 0, .••,

or aQ = bQ, a^ = b^, a^= b^, ..., a,^ = br,, •••• (216)

218. Since every finite series may be regarded as an in-
finite series in which all the terms are zero except a finite
number, the two preceding theorems apply to all series,
both finite and infinite, of the same form which satisfy the
given conditions with respect to the values of x. Thus, in
the preceding theorem, both series might be infinite or both
finite, or one finite and the other infinite. If there are
powers of x in one series which are not found in the other,
the coefficients of such powers must be equal to zero. Thus

^ aQ H- a-^x + a^x^ -\- a^x^ = ^0 + ^1^ "'" ^2^^

for all values of rr, then

a^ = Jq, a^ = ^j, ^2 = ^2' ^^^^ ^3 ~ ^-

166

COLLEGE ALGEBRA

DEVELOPMENT OF AN ALGEBRAIC FRACTION INTO A SERIES

219. We shall now use the preceding theory to show how in
certain cases an algebraic fraction can be thrown into the form
of an infinite series.

EXAMPLES

1. Develop

- into an infinite convergent series.

x^— 5 x-{- 6
Assume, if possible, that the fraction is equal to such a

series ; then we shall have

— — "*" — = a^ + a^x 4- ^2^^ + • • • + '^,._2^^~^

2/ ^~* O u/ "J" O I T—'\ I V I

Clearing of fractions, first multiplying by 6, then by — 5 re,
and lastly by a:^, we have

2 a; + 1 = 6 «() + 6 a^
— 5 an

af-^

which under the restrictions of the assumption furnishes
two series satisfying the hypothesis of 217. Therefore,
equating the coefficients of like powers of x^ we have

1

6 '

17

+ 6 «2

^2 _|_ ... _^ (3 ^^ _^

^r-2

— 5 a-^

5a,_3

+ %

••• + «;.-4

+ 6 6«,,_i

x'^-^ + 6 «,.

— 5 <x,._2

— O ^/.—-i

+ «,-3

+

a,_2

6 6^Q = 1, .*. fl^o =

6 ^j — 5 ^Q = 2, .•. 6 «j = 2 + 5 ^Q and t)^| =

79

36

6^2-5^1 + S= ^' •'• ^2 = ^'

6 a,. — 5 a^_-^ + ^;._2 =0, if r > 1, .*• a,. =

5 a.,,_^ — a,..
6

UNDETERMINED COEFFICIENTS 167

x'^—Dx-^-b o db 216 129b

values of x as make the series convergent. It will be shown
later (230) that this series is convergent if \x\<2.

Note. In practice we may write the first term by inspection, as the
ratio of the terms of the lowest degree in the numerator and the denom-
inator respectively.

S X — 2

2. Develop — — - — - into a series.

x^ — 4 a;^ -f- 2 ar

Sx-2 1 Sx-2

x'^ — 4: X^ -{- 2 X^ X^ X^ — 4: X -\-2

Sx — 2

Assume — j = «q + a-^x -j- a^x^ + ^ga;^ H \- a^x*' -\ —

As in the preceding problem, we find

and, in general, if r> 1, a''= — -^^ ^^-

Therefore, to five terms,

3 a; — 2 _ ^ x x^ o x^ 5 x^

x^-4x+2 2 2 1 1

Dividing both members by x^^ we have

3a;-2 _l_J__l_i^_5^

a;* — 1 a:^ -f- 2 a;^ x^ 2x 2 4 4

for such values of x as make the series convergent. It can
be shown that the series is convergent for all values of x
numerically between 0 and 0.5858+.

168

COLLEGE ALGEBRA

3. Develop -- — - — - — - — - into a series.

By inspection we see that the first term is - x~^. Hence,

3

since the series must be in ascending powers of x beginning
with x~^, we assume

X t: X -f- J- -t o j^ 2 I —1

a^-'Zx'^ + Sa^ ^ ^ ^ ^

4- a^^x^ 4-

Clearing of fractions,

2

~5

X -\- S a_j
-2a

-2

+

rz:^ + 3 a.

_ 9

a.

+ «_r

a:^

+ 3ai
-2a,
+ a_.

a:* H h 3 a.

z a

r-l

+ a

r-2

X'^^-^-

10

14

Therefore a_^= - —- ; a_i=- — ; a() =

81

; <^i

243'

and, in general.

^^^^a,_i^ ^^-2^^>_1.

Hence ^^-4:r + l ^1 10 14 2 46^
a^-2a^ + Sa^ S x^ d x^ 27 a; 81 243

for such values of x as make the series convergent.

UNDETERMINED COEFFICIENTS 169

220. EXAMPLES

Develop to four terms the following fractions into series
in ascending powers oi x:

^ l-\-Sx ^ 2-Sx + x'^

^ 5x^ + 2 ^ ^+8

3^—4: X^—2x-\- 4:

-3a;2+7a; . a-{-bx

o. • b. .

X -{- 4:0^ c + dx

BINOMIAL THEOREM, ANY REAL EXPONENT

221. We shall now complete the proof of the binomial
theorem as we promised in 171. We consider two cases,
according as 7i is commensurable or incommensurable.

Firsts when n is any commensurable number whatever,
positive or negative, integral or fractional.

Assume (1 + re)" = ^^ + a-^x + a^ 4- a^x^ + • • • + drX"^ + • • • (1)

Then (1 + yj' = ^o + HV + Hlf' + HV^ ^ ^ ^rV'' H •

Subtracting, and dividing by (1 + a?) — (1 + ^) = a; — y, we
have

(l4.^)n_(i^yy^^ fx^-jy\ fx]^-f\ fo^-f
(l + 2;)-(l+^) \x-yj \x-yj \x-y.

\x-y J

Taking the values of both members when y — x^ and there-
fore also 1 + ?/ = 1 + ^, we have, by 192,

n(\ + xy-^ = a^ + 2 «2^ + 3 a^x^ H (- ra^"^ H •

170 COLLEGE ALGEBRA

Multiplying both members of this equation by 1 + re,

w(l + xy = a^ + (^1 + 2 a^^x + (2 ^2 + 3 a^')x'^ + • • •

-\.[(r-l}a,_^ + ra,']x'-'^ -{-■•'. (2)
But

n(l-{- x^ = na^ + na^x + na^x'^ + • ■ + ?^(2^_Ja;^~l + • • • (3)

by equation (1).

Since things equal to the same thing are equal to each
other, the two infinite series forming the right members of
equations (2) and (3) are equal to each other for all values
of X which make both series convergent. Therefore, by
217, we may equate the coefficients as follows :

2 ^2 + ^1 = ^^^1'

3 C?3 + 2 ^2 = ^^<^*2'

ra^ + (r — 1 )c/y_i = na^_i.

But from equation (1), by ]3utting a; = 0, we find aQ = l.

mu £ nCn — V) n(n—V)(n — 2?)
Therefore a^ = n, a^ = -^ — — ^, a^ = ^ — -;^ ^,

1-2 1 • 2 • 3

and, in general, a,. = • «,._-, .

r

Changing r into r — 1 in this,

^^ - r + 2
''-1= r-l

• ^r-2?

UNDETERMINED COEFFICIENTS

171

Multiplying corresponding members of these last r + 1
equations, and canceling the common factor a^a-^a^a^-'-ar.-^^

we have . ^^^ r,. . ^x/>

__ n(n— l)(n — z) •• • (^z — r + 1 ) _ ni

~ 1.2.3-^ ~ W.

a,.

Hence our development is

(i+.)»=(«)+(»).+g)..+ ...+(»y+.... (4)

It remains to consider the convergence of this develop

ment. ^ n / s -.

n\ ( n \ n — r + \

.r-1

Since

Wy+i _ ?t — r + 1

X.

u.

Therefore

(172)
(cf. 183)

'•+1

Ur

Hence the development is convergent if la:|<l.

(206)

If

a: = - , then
a

<1; that is, |J| < la

Substituting - for x in (4), we have
a

Multiplying both members of this equation by a"^,

which is convergent provided |5| < la|.

(5)
(cf. 171, 3)

172 COLLEGE ALGEBRA

Second, when n is any incommensurable number.
Let m be a commensurable number which approaches n as
its limit. Now, by (5), if |5|< |a|,

(a + hy^ = h^ a^'^ + ("f] a^-^h + C^\ a'^-%'^ + • • •

+ h\a^-'h' + '", (6)
But L (a-\- by = (a + ^)^

m=n\ r

L i''%^-'b' = r]a^~'b^.

Denote the series in (6) by aS'^ and the series

^y« + ('!)^"-i^ + ••• 4- C^)^^-'-^'- +

by S. Then we have

L SJ = iS'„ by 190, 2.

Hence i (L S,') =L S, = S;

r=co m—n

that is, we have proved that S' approaches S as its limit as
m approaches n ; and also (a + 5)"* approaches (a + ^)" as
its limit ; but (a + 5)"" = /S" always, therefore

{a + by=-S, (190,1)

(a + J)- = {^\ a- + r^^) a^-^b 4- Q a"-2J2 + . . .

or

Another proof is given in 378.

+ f^U''"^^''+ •-.

UNDETERMINED COEFFICIENTS 173

222. EXAMPLES

Expand to five terms :

1. (x + y)^. 7. (a^-^^)?.

2. (2x + 2>y~)\ 8- (2 + 3x2)-2.

3. (3a;2+2?/3)i 9. Ux^-X^-

10.

5. (2-0.2)1. Va2 2/2

. _gs| 11. (2 a;- 3^2)-!.

^' V "^J * 12. (2:2-2^3)-|.

Develop into a series to a term containing as high as the
fourth power of x :

13. VI + 3 a:. 15. Vl + 2 a: + 3 a^.

14. V2 + 3 2;2. 16. V2-5a;2.

Develop to five terms :

17. V2 a + 3 6 in ascending powers of h.

18. ■\/2 a + 3 6 in ascending powers of a.

19. Find the coefficient of the 7th term in V3 — 2 x2.

20. Find the coefficient of x^^ in V3 — 2x^.

DECOMPOSITION OF FRACTIONS INTO PARTIAL FRACTIONS

223. In certain mathematical investigations, as in the
integral calculus, it is often required to express an algebraic
fraction as the sum of several others whose denominators
are of a simpler form than that of the given fraction. Thus

174 COLLEGE ALGEBRA

it may be required to express — — as the sum of

x^ — 0 x-\- 6

such fractions. Since the denominator is composed of the
factors (re — 2) and (:r — 3), we take these factors as the
denominators of the required fractions, and assume if pos-
sible for all values of x

2a;+l ^ A B

x^— 5 X -\-6 X — 2 X — S

where A and B are undetermined constants.
Clearing of fractions, we have

2 a; + 1 = A(x - 3) + Bi^x - 2). (1)

Equating coefficients of like powers of x, (217)

-3^-2^ = 1,
A + B=2.

Solving for A and j5, we have J. = — 5, ^ = 7, for all values

of X.

u 2a; + l 7 5
Hence — — — ^ ~ = -•

X^ — D X + b X — €) X — I

Thus our assumption is justified ; for it requires us to
find the values of two unknown quantities when two inde-
pendent equations of the first degree containing them are
given. This always gives a determinate solution for each
unknown.

We might have obtained the values of A and B by
another method. Since equation (1) is true for all values
of X. it is true when 2: = 2 and when a: = 3.

UNDETERMINED COEFFICIENTS 175

When a; = 2, (1) becomes (2 x + 1)^=2 = ^(^ — 3)^2'
whence A = l ^t^) =—5.

X — 3 J x=2

Similarly, B = ('^l^±1) = T.

X — '2 /x=3

Observing the form of A and B which we have last ob-
tained, we see that A is obtained by putting x= 2 in a
fraction which is derived from the original fraction by neg-
lecting the factor x — 2 in the denominator. Likewise for B.
And in general it may be shown that the same law for find-
ing the constant numerators always holds when the denom-
inator is a product of different linear factors. The value
of X to be used to find a required numerator is that which
makes its denominator vanish.

224. Four forms of fractions are treated according to the
nature of tlie fa,ctors of their denominators. Firsts those ivhose
denominators consist of different linear factors ; second^ those
whose denominators are poivers of linear factors ; thirds those
whose denominators consist of irreducible quadratic factors ;
fourth^ those whose denominators consist of combinations of the
factors mentioned in the first three cases.

225. Since every rational fraction whose numerator is not
of lower degree than its denominator can be reduced to the
sum of an integral expression and a fraction whose numera-
tor is of lower degree than its denominator, the decomposi-
tion of only this type of fraction is treated.

226. The forms into which the types of fraction men-
tioned in the four cases of 224 are to be decomposed into
partial fractions are as follows :

1. A fraction of the first type is decomposed into a sum
of fractions whose denominators are the several linear factors

176 COLLEGE ALGEBRA

of the original denominator and whose numerators are un-
determined constants. Thus

-r

x^-Qx^-^llx-6 (a;- l)(a;-2)(2;-8) x-1 x-2

+ — ^•
x — Z

2. When the denominator of the fraction to be decom-
posed is a power of a linear factor, the denominators of the
partial fractions are the different powers from the first up
to the power occurring in the original denominator, and the
numerators are undetermined constants. Thus

x^-\.x-^r\ A ^ B ^ 0

{x-\f {x-Vf {x-Xf x-1

The reason for this statement is evident without a formal
proof. The left-hand member may, by 85, be written in the

degree one lower in x than the original numerator. Like-
. p _0(x-l)-\-B_ B 0

^^^^ (x-iy~ (x-iy ''{x-iy x-i

3. In this case the denominators of the partial fractions
are the several quadratic factors of the original ; the numer-
ators are binomial expressions of the first degree in x, each
containing two undetermined constants. Thus

x^-\-x^-{-x-hl _Ax + B Cx-\-I>

(a;2 + 3)(rz^2_^a^+l) x^-^S x^-^x-\-l

An expression of the form x"^ -^ ax + h is equivalent to
(a; + «) (a: -F yS) where a and 13 in this case will be surds or
imaginaries, and conjugates of each other. Hence the frac-

UNDETERMINED COEFFICIENTS 177

tion whose denominator is x^ + ax + b is the sum of two

A' B'
fractions of the form 1 -, and hence must itself

X -\- a x-\- p

Ax -\- B
be of the form — — -• For a full discussion of these

x^ -\- ax -\-o
special types see 228.

4. In this case the original fraction is decomposed into
the sum of all the partial fractions to which each factor in
its denominator gives rise, as stated in the three preceding
cases. Thus

x^ + ^x-\-l _ ^1 A A

(a;- l)3(a;- 2)2(0^2^4) a;-! (2: -1)2 (x-iy

B, B, C,x+C,

x-2. Qx-2y 2:2 + 4

227. We complete the solution of the first three problems
of 226.

1. By the second method in 223,

\{x-2)(x-n)J,^,

— o

(-l)(-2)

B = l 2a^-^+3 ^ =_^ = _9,

(^_l)(^_3)y^^^ 1(_1)

^^f 2x^-x + S \ ^ 18 _g
\(x-lXx~2)J,^, (2)(1)

Hence „ 2^-^ + 3 ^^ 9_^ 9

afi-63? + nx-6 (:»-l) i^--} (^-3)

2. The second example may be solved as in the first
method of 223 by clearing of fractions and equating coeffl-

N

178

COLLEGE ALGEBRA

cients. We may also use a modification of the second
method. Thus after clearing of fractions we have

x^-hx-{-l = A + B(x-l')-\- C(x-iy.

When X = 1 this becomes A = (a;^ -i- x i- l)j;=i = 3.

Thus A is obtained by putting x = l in a fraction which is
derived from the original fraction by neglecting (a; — 1)^ in
its denominator. Thus

x"^ -{- x + 1 _

+

B

-f-

C

{x-if {x-\y (x-if x-i

Transposing, combining, and dividing out the factor x — 1,

x+2

B

+

0

(ix - 1)2 (x - 1)2 ■ ^^ _ 1

Proceeding as before, B = (x-{- 2)^.^i = 3.

Putting in this value of B, and transposing as before,

1 _ (7

Therefore,

X —\ X — 1

x^ + x+1 3

whence 0=1.

+

T,+

3.

(2; -1)3 (x-iy (x-iy x-i

a:3 + 2^2 + a: + 1 _ Ax -\- B , Cx + D

+

(a:2+3)(a:2 + a:+l) x^ + ^ x^ + x-^1

Clearing of fractions,

x^ + x'^-\-x-\-l = A
+ 0

a:3 + ^

x^+ A

+ B

+ B

+ i>

+ 3(7

X

+ B

+ 3i).

UNDETERMIXED COEFFICIENTS 179

whence

Subtracting (2) from (1),

Subtracting (3) from (2),

Adding (4) and (5),

Subtracting 2 x (6) from (7),

And by substituting, i^ = f , B = — ^, A=^.

Therefore

x^ -{- x^ + X -\- 1 _ G X — 2 X + S

A \ G =1,

0)

A^B + Z»=l,

(2)

^ + fi+3C =1,

(3)

B +8i)=l.

(■t)

- ^ + G - i> = 0.

(5)

- 8 C + 2> = 0.

(6)

C+2i) = l.

(7)

), G = f

(2:2 + 3)(2:2 + a: + l) 7(^:2 + 3) 7(^:2 + 2:4-1)

228. T/te decompositions stated in 226 <?ari he shown to he
correct hy oh serving that the numher of unknown constants
introduced is the same in each case as the numher of equations
arising hy equating coefficients for the solution for the unkriown
constants^ and that the solutionis are in each case unique.

If the denominator of the original fraction is of the degree
?^, each of the various cases will give rise to n undetermined
constants. When the partial fractions have been reduced
to a common denominator and added, the numerator will be
of the (>i— l)th degree, thus giving rise to n equations for
the determination of the n constants. But the agreement of
the number of constants and the number of equations is not
sufficient to show that the solutions will be unique. For

this purpose we consider the fraction ^ — -^ where /(a:) is

of degree w, and <^(x^ of degree (n — 1). Let fi^x^ contain
r different linear factors (x— «j), (x—a^^ •••, (a: — a^);

180

COLLEGE ALGEBRA

s sets of factors (x — P^\ (^x— P^\ •••, (x— ^sYs], and
t sets of factors (^^ + 7i^), (^'^ + 72^)? ••> C^^ + T^^.*
Then, by 226, we ought to have

+ ••• +

^,

+

B,

B.

x-ar (x-p^y^ (a;-/3i)'i-i

+

X— /3i

+

+

+

^1

+

(:.-yS,y. (x-^,ys-^

+

X,

^- A

H » H ^^ ^

x^ + Ti''^
+ ••• +

a;2 + 72^
^ + 7<^

Here we have r-{-r^-{-r^+ • • • + r^ 4- 2 ^ = 7^ constants. The
second method of 223 and the method of Example 2 of 227
show that the ^'s, -6's, •••, Z's can be determined uniquely.
As to the (7's and D's, instead of the fractions of the form

— '^ — , we might have taken fractions of the form
Q^ + 7^

R,

s.

-.+

XLr

rr + 7l^ x — ^^% x -\- y^i x — y^z

-.+

^o

-,+ ••• +

Bt

^t

a; + 7i^ x— 7^1

by the first case, where the M's and S's would now be among
the A's and therefore could be uniquely determined.

Each pair like M^ and aS'j would be complex conjugates, for

''•=[/g)^^^^^^'^'

, S^ =

Vi*

Mx)

L/(^)

(x - 7i0

=Vi»

* Since any quadratic expression ax"^ -\- bx -{- c can be thrown into the

form f Vax + — ^ ) + c — — , the treatment of factors of this type is con-

V 2Va/ 4a

tained in that here given.

UNDETERMINED COEFFICIENTS 181

Put f(x) = [^|r (x)] (2: + 7i0 (x - 7^0 .

Then E = ^("^lO S = ^CtO .

' 2[^/.(-7i0](-7i0' ' 2[^/.(7i0](7i0'

that is, i^i and ^j come from each other by changing i into
— i in the function which expresses either. Hence, by 118,
H^ and jS-^ are conjugates and hence of the form a + bi and

a — ^^.

TT M. , )S. a-\-hi , a — hi 2(ax -{- hy.^
Hence ^ H ^ = — ^ . H . = -^— fLZ .

a; -f 7jZ 2: — 7J^ 2: + 7i^ ^ — 7i^ ^ + 7i

Whence Cj = 2 «, i)i = 2 by^

In like manner similar conclusions c
regard to any other proposed type, including

In like manner similar conclusions can be obtained in

Ux-{-F

229. EXAMPLES

Decompose into partial fractions :

^ 4:x^-59x-29 5r^-{-6x^+5x

Qx-{-2){x-b)QSx + iy ' {x^-l)(x-^l)

2. . ^ + ^ 8. ^ + ^

3

{x^ + 1) (a; + 2) 15 x-^ 2x^-0^

7rr3-15a;4-12 22:-l

{x-\-l)Xx-iy' ^'

a;2 — 7 a; + 12
2a:4_3^^7

a;4-l ' ' (2:2+l)(a;2 + 2'H-l)

* (2a;-l)(a: + 2)2 * x^ -{- x^ - 2

(5x

+ 7)c3:.-2)(4-
a:2+l

-^)

(X-

1

182 COLLEGE ALGEBRA

13. _^izLi_. 17. 6x^-\-x-l

x^ + x^+1 (2^2 + IX^ - 2)(2: + 3)

14. J,„ ... 18. ^+1

15.

{x - l)2(a;2 + 1)2 (:c2 _^ i)2(^ _ i^^

^6 _ 1 2:4-1

x^-x-^1 x^ + 2x^-2

^* (^^-iy(^x-2){x^ + l) ' x^ + x^-2

GENERAL TERM IN THE DEVELOPMENT OF AN ALGEBRAIC

FRACTION

2 2; + 1
230. In 219 we have developed "" into a series,

X^ — 0 X -^ D

and have given the law of relation connecting any three
consecutive coefficients. We shall now obtain a single
explicit expression for any coefficient independently of the
other coefficients.

By 223,

22:+l ^ 7 5___5 7_

a;2— 5a? + 6 x — o x — 2 2 — x S — x
= 5(2-0:)-!- 7(3 -2:)-i,

which we can expand by the binomial theorem into a con-
vergent series provided |rr| < 2.
We have

^^ + ^ =5(2-1+2-22;+ ... +2-i-^x^^+ ...)

X^ — b X -\- 6

_ 7(3-1+ 3-2 ^^_ ... _^3-i-.^.r^ ...).

Hence the coefficient of x'' or a,.

5 7

= 5.2-1-^-7.3-1-^ =

2^+1 3''+i
for any value of r.

UNDETERMINED COEFFICIENTS 188

231. EXAMPLES

1. Use the formula of 230 to obtain the first five terms

of the development of — -^ — — into a series.

x^ — o X -\- 0

Find the general term in, and obtain five terms of the

development of :

^ X -{-1 S X

x^-Sx + 2 x^-2x-{-l

SUMMATION OF INTEGRAL SERIES

232. We shall use the method of undetermined coefficients
to obtain the sum of 7i terms of a series whose nth term is of
the type , ^_,

Thus to find the sum of the first n terms of the series

12 _j_ 22 _|_ 32 _|_ ... _|_ ^^2^ ^yg assume

12 + 92 + ... + ?i2 = ^ + Bn + (7n2 + Dti^ + IJn^ + ••• (1)

as true for all values of n. It is therefore true for the next
higher value of 7i.

Hence I2 + 22 + 32 + • • • + «2 _^ ^^^ ^iy = j_-\- B(n + 1)

+ C(n + 1)2 + I)(7i + 1)3 + ^(n +iy+ •••. (2)

Subtracting (1) from (2),

(^i + 1)2 = ^ _^ (7(2 n + 1) + i>(3 ?i2 + .3 n + 1)

+ ^(4>i3 + 6m2 + 4>/ + 1)+ •••.

Since this is an identity in which both members are finite
series, the degree of both must be the same, that is, the second
degree, whence it appears that all coefficients like U, F^ •••

184 COLLEGE ALGEBRA

must be equal to zero. Equating the coefficients of the
corresponding powers of n in the remaining terms, we have

B-{- C+ D = l,

2 (7 + 3 D = 2,

3i) = l,

whence -^ = 3' ^ = 2 ' ■^ — \'

Substituting, I2 + 22 + 32 + ... + /i2 = ^ + l ^ + 1 ^2 + 1 ^3,

Now putting n = 1, we have 1 = J. + 1 ; therefore ^ = 0.
Reducing to a common denominator and factoring,

12 + 22+32+ ... +ri2 = ^(2nH-l)(n + l).

b

Similarly the sum of n terms of any other series of similar
type can be obtained. It is to be noted that the series to
be assumed as the sum of n terms is to be of a degree one
higher than that of the nth. term of the given series.

Thus

1.2.3 + 2.3.4+ ... +m(72 + 1)(^ + 2)

= A + Bn-\- Cn^ + Dn^ + EnK

233. EXAMPLES

Find the sum of n terms of :

1. I .2 + 2.3 + 3 .4+ ....

2. 1.2.3 + 2.3.4 + 3.4.5+ ....

3. 13 + 23+33+.... 5. 12 + 32 + 52+....

4. 14 + 24 + 34+.... 6. 1.22+2.42+3.62+....

7. 1 .2.4 + 2.3.5 + 3.4.6+ ....

8. The series whose nth. term is 2 n^ + 3 /i + 1.

CHAPTER XIII

CONTINUED FRACTIONS

234. Definition. An expression of the form

h

a -f-

d

e^ ^

9+"-

which is for convenience tvritten in the form

h d f

a + - - -•••
^+ e + g

is called a continued fraction.

Continued fractions fall into two classes, those which te?'-
minate and those which do not. The latter class is called
infinite continued fractions. Of the first class we shall con-
sider here only those which are in the form

111 1.

^ a^-\- a^-\- a^+ + a„

A more convenient notation for this is (a^, a^^ ••-,«„). If
- denotes the value of the continued fraction, we have

0

. . V

235. Theorem. It is possible to express any fraction — > P
and q being integers^ as a continued fraction.

185

186 COLLEGE ALGEBRA

7) V 1

For, ^ = a. -\- -^ = a^-{- - '

q ^ q q

Continuing this, we have

^71

1

Li

1

1

^2 +

a,

1

«2, •••,«„).

It is to be observed that the process here used is the same
as that employed for finding the highest common factor of
the two numbers p and q and therefore will terminate,

The student must not fail to observe that when p <q, a^ =
0, and rj = p.

EXAMPLES

1. Reduce |J| to a continued fraction.
The work is as follows:

629)271(0
000

128
117

271)529(1
271

11)13(1
11

258)271(1

258

2)11(5
10

13)258(19
13

128

1)2(2
2
0

Hence aj = 0, a^= 1, a^ = 1, ^^ = 19, a^ = 1, a^ = 5, ^7 = 2,
and we have ||i = (0, 1, 1, 19, 1, 5, 2).

CONTINUED FRACTIONS 187

2. Reduce ^l to a continued fraction.

Proceeding as in Example 1, Ave find J|| = (2, 4, 3, 1, 18).

236. Let ^ be denoted by ll,
1 9i

l^a,^,+ l g

^1 + T- = ' ' ^ ^^ ^ by ^-^\ etc.,

where in each case the ^'s denote the numerators, and the ^'s
the denominators, then we define li, 1^, li, etc., as the/rs^,

second^ thirds etc., convergents of the continued fraction.

237. The law of formation of these convergents may be
found inductively thus :

p^ a^a^a^ + <^3 + «i ^^zPi'^ Pi

P4^^_4PS±P2 g^^

Let m be a value for which this law has been observed to hold.

rpi Pm ^mPm — l > Pm—2

9m ^m9m-\ ~r 9m- 2

Now -^^ is obtained from — by substituting a„^ -\ in

9m+\ (j[m ^m+\

place of a^. / -j n

I a^ -\ j ]^m-\ + Pm-2

ence = y :j — r

^m + l\^mPm-l i pm-l) + Pm-\ ^m+\Pm+ Pm-\

^m+l(^^m9m-l + 9m-2) + 9m-l ^m+\9m ' 9m-\

188 COLLEGE ALGEBRA

Therefore the law holds for -^^, and since it holds for m

= 3, it holds for m = 4, then 5, and so on, and thus it holds
for all applicable integral values of m greater than two.

Hence the theorem Pni = ^7nPm-i+ Pm-2^ a-nd qm = cCm<im-i
H-^m-2 when m>2.

238. EXAMPLES

1. Find the successive convergents to ||^.

We have found in Example 1, 235, that a^ = 0, ^2 = 1, a^ = 1,
a^ = 19, a^ = 1, ag = 5, a^ = 2. Hence the convergents are
^, h h ti It' iff' fl4- The whole work is checked by
finding the last convergent equal to the original fraction.

2. Find the successive convergents to J^|.

3. Find the successive convergents to ||^.

4. Convert -^Jj- into a continued fraction, and check the
work.

5. Convert ^^j^- into a continued fraction, and prove the
correctness of the work.

239. Definition. A recurring continued fraction is a non-
terminatiyig continued fraction in which from a certain point on
a number of quotients periodically recur. Thus

54-1 11111

"^4 + 3 + 2 + 7 + 2 + 7+*"

A 1111111

3 + 4 + 3 + 4+3 + 4 + 3 + '"

are examples of recurring continued fractions. In the second
the quotients recur from the beginning.

240. Theorem. Every recurring continued fraction is a root
of a quadratic equation.

CONTINUED FRACTIONS 189

Let the fraction be
« +1 ... i 1 1 ... 1 1 1 1 ..

or, as it may be written, (a^, a^, •••, a,., 5j, h^, •••, 6^).

Let X denote the entire continued fraction, and y denote the
recurring portion,

6+111 1

1 hen x= a^-] — — ••• — — .

^2 4- «3 + +a^-\-y

Then, by 237, x=^f±^.

p

where — ^ is the rth convergent of x.

Hence y=&L±Ilzl,

and ^,?/2 + (g,_^ -Ps^y- Ps- 1 = 0,

where ^^ is the sth convergent of y.

Substituting the value of y in terms of x as obtained above
in this equation, and clearing of fractions,

qs{Pr-l - Qr-l^y + fe-1 -PsXPr-1 " Q.-l^XQr^ " ^r)

190 COLLEGE ALGEBRA

or [^.^r-l^ + iPs - qs-l) QrQr-1 - Ps-lQ>^]^'^ - [2 qs^r-lQr-l
+ iPs-qs-l){Pr-lQr+PrQr-l)-^Ps-lPrQr']x
+ qsPr-l + {.Ps - qs-l)PrPr-l " Ps-lP,^ = ^'

Example. Find the quadratic equation of which

1111

3 + 4 + 3 + 4+'"
or (0, 3, 4) is a root.

Let - equal the value of the fraction.

y

rpi 1111

1 hen - = - — - .

y 3+4+^

Here iLi = _; -Li = — .

qi 3 ^2 13

Whence - = — ^—^ — , or if x= -^

y 13j/+3 y

3 0^2+12 2: -4=0.

c 1 • +!.• +• ±4V3-6
bolvmg this equation, x = .

3
Since the continued fraction is manifestly positive, its value
. 4V8-6

241. EXAMPLES

Find the quadratic equations of which the following are
roots, and exj)ress these roots in surd form:

1. (0, 2, 5). 3. (1, 2, 3).

2. (0, 3, 2). 4. (3, 5, 2).

Suggestion. Put x — S = y.

5. (2, 5, 7). 6. (1, 2, 3, 4, 5).

CONTINUED FRACTIONS 191

242. It may be proved tliat every jwsitive quadratic surd, or
any binomial quadratic surd in ivhicli one term is rational may
he expressed as a recurring continued fraction.

EXAMPLES

1. Reduce V5 to a continued fraction.

The greatest integer in V5 is 2.

Hence we write V5 = 2 + ( V5 — 2) = 2 +

V5 + 2
Again, the greatest integer in Vo + 2 is 4.

Hence V5 + 2 = 4 + ( V5 - 2) = 4 + —J — ,

V5 4-2

where the fraction to be further converted is the same as
before.

Hence V5 = 2-hi: i ...=(2,4).

4 + 4 +

As a verification of the results we employ the methods

°''"'- ^ o_ 1 _ 1

4 + (2:- 2) x^-2

Hence x^= 5, and x=± V5»

and X = Vo is a root of the equation.

2. Reduce VT to a continued fraction.

VT = 2 + ( VT - 2) = 2 + —S — = 2 +

1

V7 + 2 V7 + 2

3

3 3(V7 + 1) V7 + 1

192 COLLEGE ALGEBRA

V7 + 1 , , V7-1 . , 6 ,1

— - — = lH — = lH . = iH ^

2 2 ^2(V7-1) V7 + 1

3
V7 + l^-|+V7-2^-^ , 3 ^1, 1

3 3 3(V7 + 2) V7 + 2

V7 + 2 = 44-(V7-2).
Therefore VT = (2, 1, 1, 1, 4).

Let the student verify the result.

3. Reduce V47 to a continued fraction.
V47 = 6 + (V47-6)=6 +

V47 + 6

11

V47 + 6 . V47-5 , 1

11 11 V47 + 5

2

V47 + 5 . V47-5 . 1

11

V47 + 5_^ , V47-6_^ , 1

11 11 V47 + 6

V47 + 6=12 + (V47-6).

Hence V47 = (6, i, 5, 1, 12).

It will be observed that the last quotient before the period
begins to recur is twice the initial quotient, and that within
the period there is a reverse recurrence of the quotients. A
general proof of these properties could be given. They are
of value in checking the accuracy of the computation.

CONTINUED FRACTIONS 193

Convert the following quadratic surds into continued
fractions, and verify the results :

4. Vn. 6. V21. 8. VT9.

5. Vl7. 7. Vl8. 9. 2-fV3.

10. Show that 2 - V8 = (0, 3, 1, 2).

Convert the following surds into continued fractions, and
verify the results :

11. 5-V2. 13. 7-V7. 15. ^^11^.

5

14. 2+V3 7_V6

lb.

12. 3-V5. ^«.

^ 3

243. Forming the differences

^ — ^= , etc.,

(236)

and denoting by m that value for which this has been ob-
served to hold, we have

Pm Pm-l^j-'^T _
9.m 9m -I 9.m-\2m

•g^^ Pm_ _ Pm-1 __ Pm9m-l ~ Vm-l^m ^

9.m 9.m-\ %n-\9.m

Therefore the numerators of the second members must be
equal, or _ —(_\yn

Thpn Pm-^\ Pm ^m+-[2^m "^ Pm-\ Pjn Pm-\ ^w Pm9.m-\

9.m+l %n ^m+l9m "r 9m-l Qm ^m^m+l

= — ^ ^. (237)

9m9m+-i

194 COLLEGE ALGEBRA

Therefore P'n+i9m- Pmq>n+i ^ (-1^^^ ^

Hence the law holds for a value m + 1. Since it holds for
m = 3, it holds for w = 4, and so on for all positive integral
values. Hence the law Pm9.m-\— Pm-i%n= (~1)'" is true in
general.

244. Every convergent — is in its lowest terms, for

and therefore p,^^ and q,^^ have no common ^factor but unity.

245. Conver gents are alternately greater and less than the con-
tinued fraction, and each one is nearer to it than the preceding.
The first convergent a^ is too small, the second convergent

a^-\ — is too large because a^ is too small, the third con-
H \ 1

vergfent <^o H t is too small because ao is too small, or —

^ 'i' ' \ ^ a^

^2 + -

is too large, and — is therefore too small, and so on.

«2 + —

P

Denoting — by x,

H

X

Pm ^ -gpm+i -f- P,n Pm ^ -^( - ^Y^^
<lm Kqm^^-[-q„, q„, qm(^q7n+l + qmY

where K= a.,,^^-^ +

1 1 1

X —

^m+2 ^'^in+S ' ' ^n

Pm^-\ Pm^m+l Pm+l9m \ -'■/

qm+1 qm+ii^qm+1+qm) qm+iiKqm+i + qm)

CONTINUED FRACTIONS 195

These relations show that two consecutive convergents lie on
opposite sides of the value of the continued fraction, and
since K is not less than 1, and q,^^^ = a,^^^q„, + q,n-i>qm. it is
clear that the second difference is numerically less than the

first, and hence ^^ is nearer to x than — •

9'm+l qm

The error in taking — for x is

^<^-l)"'"' ... (-1)

WJ + 1

or

qm(J^qm+i-\-qny ^, (^ _,q,n

q»i[qm+i -r -^

Since — > — —

qm qmqm+i

11 11
we have -- > > e^> - — >

qn? qviqm+1 "" qmOim+i + qm} ^q, ^

[m+l

where e,^ = — y denotes the numerical value of the

error. — - and are looser limits^ and and

qm ^ qm+i qmq>7i+i

are closer limits of error.

qm{qm+i + qm)

EXAMPLES

1. Find the closer limits of error in taking its 5th conver-
gent for ||l

The 5th convergent is |1, qQ= 244.

Hence the error lies between — — and

41 x244 41(244+41)'

or between and

10004 11685

2. Find the closer limits of error in taking the 5th conver-
gent for f Jf f .

CHAPTER XIV

INTEGRAL SOLUTIONS OF INDETERMINATE EQUATIONS OF

THE FIRST DEGREE

246. Since by 243 of the previous chapter, Pn<ln-i— Pn-\%
= (—!)% it is easily seen that solutions of the equation
ax-\- by = \ can be found when a and h are prime to each other
if we put h =Pn-> ^ = 5'n ii^ ^^e preceding relation and choose for
X and y the numerical values of j9„_j and qj^_^ with such signs
as shall satisfy the equation ax-\-hy— 1. jt?„_j and qn_^ are
the terms of the next to the last convergent in the process of

converting - into a continued fraction. If x^ and y-^ denote
a

the solutions so obtained^ the general solution^ that is, formulas
which contain all integral solutions, are evidently

x = x^-\- rb, yz=z y^ — ra,

where r is any positive or negative integer or zero. For, we

have 7 -, 1 7 -I

ax-^ + oy-^ = 1, and. ax -\- by = 1,

and therefore a(x-^ — x^-\- b(y^ — ^) = 0,

or

iCj — a; — b

y\-y ^

whence x^ — x= — rb, and y^ — y = ra,

or x = x■^^ + rb, and y — y^ — ra.

100

INTEGRAL SOLUTIONS 197

Example. Solve the equation llx +11 1/ = 1.

17 12

We find — = (1, 1,1? 5), and the convergents are -, -^
3 ^ 11 11'

2' 11'

Therefore if we choose x = — S, and ^ = 2, the equation is
satisfied. Then the general solution is x = —S + llr and
^ = 2-llr.

247. Theorem. If x^ and y^ constitute a particular solution
of the equation ax-{-bi/= 1, theii cx^ and cy-^ constitute a par-
ticular solution of the equation ax+hy — c, and the general
solution of this equatio7i is

X = cx^ + ^^ ciyid y = cy-^ — ar.

If a and h have the common factor t?, that is, if a= da\ and h

= dh\ then a'x -\-b'y = - and therefore no integral solutions of

d

this equation are possible, since the right-hand side is a
fraction.

EXAMPLES

1. Find the general solution in integers of 13a; — 19^ = 1.

19 1 3 19

We have — = (1, 2, 6), with convergents j' 9' tTj' Hence

particular values of x and y are 3 and 2, and the general solu-
a; = 3 + 19 r and y = 2 + 13 r.

2. Find the integral values of x and y which satisfy the
equation bx-]- S y = 37.

Q 1 '^ 3 8

-= (1, 1, 1, 2) ; the convergents are -, ^, -, -,
5 1 1 J o

therefore for the equation 5 a; -f 8 ^ = 1, a:^ = — 3, y^ = 2.

cx^= —111, cy-^=14:.

198 COLLEGE ALGEBRA

Hence the general solntion is

a;=-lll4-8r=l-112 + 8r=l- 8(14 - r),
^ = 74 - 5 r = 4 -1- 70 - 5 r = 4 + 5 (14 - r),

or, denoting 14 — r by s, x=l — Ss^

«/ = 4 + 5 s,

where s is zero, or any positive or negative integer. We note
that the equation has but one solution in positive integers,
namely, for s = 0. It may also be observed that if we can
see by inspection any special solution of our equation, we can
at once write down the general solution. Thus in the pre-
ceding example, if we had seen that 1 and 4 were values of
X and ^ satisfying the equation, then the general solution
could have been written down at once.

248. EXAMPLES

Find the general integral solutions of the following equa-
tions :

1. 53 a: -19^ = 1. 5. 7 a; + 9?/ = 4.

2. 23 a; +5?/ = 1. 6. 5 a; -11?/ = 27.

3. 37a:-29y = 19. 7. 14a;-33?/ = 49.

4. 11 2; + 41 ^ = 17. 8. 3 a: -25?/ =61.

9. Two bells begin to ring together. One rings eleven
times in five minutes, and the other thirteen times in seven
minutes. What strokes most nearly coincide in the first
fifteen minutes?

Note. This problem is one illustrating the use of convergents and
not indeterminate equations.

CHAPTER XV
SUMMATION OF SERIES

249. We have already seen how to find the sum of certain
types of series, namely, arithmetical and geometrical pro-
gression and some simple integral series such as those of
232, 233.

250. Let us start with the series whose nth term is

UJ^= (^a -\- nh^ [a + (?^ + l)i] • • • [a -f (^i + m — 1)^] ,

where it is to be observed that the m factors of any term
are in arithmetical progression, and where the correspond-
ing factors of any m consecutive terms of the series are con-
secutive terms of the same arithmetical progression. The
sum of n terms of this series may be found as follows :

'n'

Let v„ = [a + (?i + m)J]w,

then v^_j = [a + (w + yn — l)J]?^„_-j = \_a-\- (n— l)5]w„.

Hence v^^ — i;„_j = (m 4- l)5w„,

and changing n into ?i — 1, n — 2, etc.,

^2 — ^1= 0^^+ 1)^"2'

^1 ~ ^'o — 0^^ + lyhu^.
199

200 COLLEGE ALGEBRA

Adding the corresponding members of these equations,
we ave ^^ _ ^^ ^ ^^^ ^i^^(^u^-}- u^ + u^-h "• + w„)

n

where 2 Uy = u-^^ -\- u^ + u^ -\- • • • + w„,

which is the sum of the expressions of the form Uj. as r
assumes all positive integral values from 1 to 7i, inclusive.

Therefore ^u = ^» ~ '^o

1 '' (m-\- 1)6

EXAMPLES

1. Find the sum of n terms of the series

1.2 + 2. 3 + 3. 4+---.
Here the nth term u^ = n(^7i + 1).

Hence zUj. = — — — — = -n{n + l)(n-{-2).

1 3 3

2. Find the sum of n terms of the series

3.5 + 5.7 + 7.9 + ....
Suggestion :
w„ = (l + 2n)[l+2(n+l)] = (2 n + 1)(2 n + 3).
y„ = (1 + 2 n)[l + 2(n + 1)] [1 + 2(n + 2)] = (2n + l)(2n + 3)(2n + 5).

251. Let us now consider the case when the nth. term is
the reciprocal of that considered in 250, viz.,

1

" (a + n^>)[a+ {n + l)b] .-.[«+ (71 + 771 - 1)5]

Let v^^ = (a + 7ih')7i„^

then v.n_^ = [a + (?i — l)5]i/,,^_i = [« + (n + m — l)5]t^„.

Hence v,^ — i'„_j = — (m — T)hu,^.

SUMMATION OF SERIES

201

Changing n into n—1, 7i — 2, etc.,

v^ — Vq= — (m — l^hu^.

Adding the corresponding members of the several equations,

or

1 (772-1)6

Ur

Note. It is to be observed that this formuLa for the sum of n terms is
the same as that of the preceding section when 7n is changed into —m, but
it is not to be supposed that one series can be obtained from the other by
changing m into — m. In fact in each series 7n must be a positive integer.

252. EXAMPLES

1. Given u„ = n(7i -\- l)(w + 2). Find the sum of n terms.

2. Given u,^ = n(ii + 2)(?i + 4). Find the sum of n terms.
Suggestion : u^ = ?<[(« + 1) + 1] \_{n + 2) + 2]

= ;;[(n + !)(« + 2) + 2(n + 1) + (n + 2) + 2]
= n(n + 1) (n + 2) + 3 7i(n + 1) + 3 n.

Thus iij^ is resolved into three parts, each of which can be
treated by 250.

Find the sum of n terms of each of the following series :

3. 1.2.4 + 2.3-5 + 8.4.6+-.-. .

4. 1.4. 7 + 4. 7-10 + 7. 10.13H-....

5. 1.7 + 2.8+3.9+....

6. 1.7 + 3.9 + 5.11+.... 8. I2 + 22+.32 + ....

7. 1.22+3.42+5.02+.... 9. 124-32 + 52+...,

202 COLLEGE ALGEBRA

10. Show that 12 - 22 + 32 - 42 -f ... + (2 w + 1)2

= (7i + l)(2n + l).

11. Show that 12 _ 22 + 32 - 42 + (2 n)'^

= — w(2 n + 1).

Find the sum of n terms of each of the following series :

12. 13 + 23 + 33+....

Suggestion: n^=n(n + l)(n + 2) - 3 w(n + 1) + n.

13. 14 + 24 + 34+ ....

14. _l__+_J_+^+....

1.2-3 2.3.4 3-4-5

15. — = 1 = 1 \ .

1.3-5 3.5.7 5.7-9

16. \ \ = \ .

1.2.42.3.53.4-7

Suggestion : w„ — — ^

n {n + 1) (n + 3) n (n + 1) {n + 2) (n + 3)

(n + l)(/i + 2)(n + 3) n(n + l)(n + 2)(n + 3) *

17. +— — ^ + + ....

1-3. 5. 73. 5. 7-95-7-9- 11

Suggestion: n^ = i(2 w - l)(2n + 1) + ^.

18. = 1 = 1 1 .

1-2-5 2-3-0 3-4-7

19. Find the number of shot in a pyramid with a trian-
gular base with 40 shot on a side.

20. Find the number of shot in a pyramid having a square
base with 50 on a side.

SUMMATION OF SERIES 203

21. Find the number of shot in a wedge-shaped pile with
a rectangular base, the lower layer containing 21 on one side
and 14 on the other, the next layer with 20 on one side and
13 on the other, the top layer being a single row.

22. Find the number of shot in a wedge-shaped pile of the
same sort, of which the lower layer contains m shot on one
side and n on the other, where m is greater than n.

RECURRING SERIES

253. Definitions. A recurring series is a series in which, be-
ginning ivith and after a certain term, each term is equal to the
sum of a fixed number of the 2^receding terms multiplied re-
spectively by certain constants. A recurring series is of the
first, second, third, etc., order according as the fixed number
of terms is one, two, three, etc. Thus in the series

l^^x^^x^ + 4:X^-\ hW2;"-iH ,

naf-i = 2 x(^n - l).T"-2 - x\n - 2)a:"-3,

or nx""-^ — 2 x(n - l^x"'^ + a^(n — 2)x''-^ = 0, if n > 2,

and the series is of the second order. In this relation the
multipliers which are constant with respect to n are 1, — 2x,
and x^. Their sum, that is, the expression 1 — 2x-{-x^, is
called the scale of relation for the series and in general if the
Uq-{- u^x -j- ti^x"^ -{-•■•+ iiyiX^ + ••• is a recurring series of order
r, and if

UnX"^ ■{- P'^XUn-iX^~'^ + j02^W'«-2^"~^ + h i?r^'''?^n-r^" ~ '" = ^»

the expression 1 -^ p^x-\-p^x^-\- ••• + p^x^ i^ the scale of relation
for the series. When the constants p^, p^,---,p,., are known,
that is, when the scale of relation is known, the series is
determined.

204

COLLEGE ALGEBRA

254. If we are given any 2 r consecutive terms of a series
of order r, we can find the scale of relation, because we are
able to form r independent equations between the r constants
of the scale. Thus to find the scale of relation of the series
of the second order,

we have 11 x^ -{- 'px %x-\- qjr = 0,

and 43 x^ -\- px 11 x^ + qx^ 3 = 0,

or 11-{_3^_I_,^^0, (1)

and 43 + lljt? + 3^=:0. (2)

Subtracting 3 times (1) from (2), 10 + 2 |? = 0,
or p = — 5^

and therefore, ^ = 4.

Hence the scale of relation is 1 — 5 x + 4: x^. By using this
we can determine as many more of the terms of the series
as we please. Thus

UnX^ — 5 xUn^-^x'^~'^ + 4 x^u^i_2X^'~'^ = 0,

or w„ = 5 Un_-^ — 4 ^t,^_2, Avhen 71 >1.

Hence, u^ = 215 — 44 = 171,

^,^ = 855-172 = 683,

u^ = 3415 _ 684 = 2731,

and the next three terms are 171 x^^ 683 x^^ 2731 x^.

255. To find the sum of n terms of a recurring series we
proceed as follows : Taking the series of order two,

6^ — Un ~p U-l X ~\~ ttnX ~\~

~r U^i_-^x ,

(l;

pX8,, = PUqX + pU^x"^ + • • • + pUn-2^''~^ + pUn-i^'^^

qx^s^ ■— qu^x^ -f-

SUMMATION OF SERIES

205

Adding, we liave
(1 -\- 2)x -{- qx^)s,^

since u,.-\- 2^^,-1 + ?^;-2 — ^» when ?- > 1.

Hence,

_ '?<Q + (u^ + /;^o)a; + (pun~i + qif'n-2)^'" + g^n-r^""^^

(2)

256. For values of x which make the series convergent, it
is necessary that L u^x^ =^^ by 196, Cor. 1. Taking the

72=00

limits as n becomes indefinitely great of both members of

Equation (z), we have s = -^ — ^^^^ — ^ 1^, since M„_ja;\

Un-2^^\ Un-yc'^'^^^ may be written xUn-iX^~^, x^Un_2X^~^^ x^Un_^x^~'^^
respectively.

257. Ihe expression -^^-:^ — - — ^ ^ is the generating

function of the series. This is obvious because it is tlie sum
of the series, and the series may be reproduced as follows :

2:" + ••-.

1 -\-px-\- qx^ "^

- u-iUy -\- v.-

yt T •■

• T- ^)i^ ^

Then

u^ + (pi(o + yi)x = v^ + ^1

x-h V^

a;2+ .

•• + Vn

+ 7^i'o

+ pv^

+ PVn-l

+ ro

+ qVn-2

Equating coefifiicients, Vq = Uq, pv^ -\- 1\ = jj»?^q + i^j, and z'j = Wj,
V,. +pv^_j + (yi'^_2 = 0, r>l, and this is the relation which
gave the successive terms beyond u^ in the original series.

206 COLLEGE ALGEBRA

258. EXAMPLES

1. Find the scale of relation and the generating function
of the series 1 -\- 2 x -{- 5 x^ -j- SI x^ -\- •••.

Find the generating function of each of the following

series :

2. 2-{-9x-\-25x'^ + 66a^-\- --'.

„ 5 , 3a; 21 x'^ , 45 x^ .

4. -S-^-Sx-lSx^-i-lSx^- '--.

5. 2-9x-{-SSx^-lllx^+ -".

6. S — X — 4:X^-\-Sx'^ — X^— •■■.

7. 1- Sx-{- 6x^-10:}^ + 15 x^- 21 x^+ '".

8. 1 + 22 2: + 32 a;2 + 42 x^ + 52 x^ + (j^ x^ + ....

9. l-Sx-]-5x^-7x^+ ....

10. Find the nth term of the series

_3_11U_17£^_39^_

2 4 8 16

11. Find the 71th term of the series

3 _ 19^ 113^ _ 679^

2 12 72 432 "'*

12. Find the sum of n terms and the generating function
of the series Uq + u-^x + u^x'^ + ••• + Un_-^x^~'^ + •••? whose scale
of relation is 1 -{- p^x -{- p<^x^ -{- p^oi^.

FINITE DIFFERENCES

259. If the terms of a series are iIq, u-^^ u^, 7/3, .••, u„^ ...,

then let A?/q, A?/j, H^u^, •••, Ai6„_j, •••, denote the differences

SUMMATION OF SERIES 207

A^Wq, A^z/j, ^^u^, '••, denote the differences
Awj — Awq, Awg — Amj, Ai^3 — Aw2,
and in general let

A^?fo=A^-i?^i-A^-iMo, A^i^i = A''-iw2 - ^"""^^^ ••••

The differences A?*^, A^?/;^, A'^%., •••, A^%., are called the differ-
ences of the firsts second^ third, •••, rth order respectively.
A may be considered as an operator, and that it is distribu-
tive is seen from the following:

= U^+^ — lij. + llk+i — Uk = ^Ur + A%.

Also A(Aw,) = A^w,.

260. Having given a number of terms, we may, by find-
ing the differences, find the general term.

By 259 we have Uj,+i = Ur + Aw^.
Thus, Uj^ = Wq + Auq,

u^ = Ui H- Au^ = i<Q + Ai^Q + ^('^0 + ^'^o)
= Hq + Auq + Ai<j) + A\,
or ^2 = ?^Q + 2 A?<^^ + A^z^Q.

Let m be a value for which it has been observed that the
following law is true :

0 ' \ -1 j'-"0

0

_L / "Maw-,/ .

m

+ ( "' W%

208 COLLEGE ALGEBRA

Since u^+i = w^^ + ^^m^ we have

»«.: = (;)"o +(7) A»„ +g)A^«„ + ... + (-)a-.„ + ...

\m — 1/ V^^i/

or, remembering that f ) + ( -, ) — ( ) ^^^^^ ^^^^^ ( j

0 / \7nJ W + iy

'm + 1

and hence, by the principle of mathematical induction, the
law is true in general.

We may write the above symbolically in the form

u,, = (1 + AT'Uo,
which gives the (w-f-l)th term.

261. The sum of m terms may be found as follows :

We have v^^ = ?/q,

u^ = u^ + Auq,

u^ = Mq + 2 A?/q + A^Wq.

Hence, Sg = k.^ + ?/i + ti2 = 3 iCq -\- 3 A?^^ + A^Uq.

SUMMATION OF SERIES 209

Let m be any value for which it has been verified that

«».=(7)«o+©^»o+-+(:)a-x+-+(:)a»-%„.

Then , . . . ,

+

Therefore the law holds for a value of m one greater. Since
it is true when m = 3, it is true for m = 4, 5, •••, that is, for
all values of m. We may write symbolically,

8 m

A

%•

262. EXAMPLES

1. Find the wth term and the sum of m terms of the
^^^^^^ 2 + 2+8 + 20 + 38 + ....

Finding the differences of the various orders, we have

2 2 8 20 38

0 6 12 18

Q Q Q

0 0

That is, Wq = 2, A?(,3 = 0, Ahi^ = 6, A% = 0 when r > 2.

Note. That A'«o = 0 for all values of r greater than 2 follows from the
assumption that a sufficient number of terms have been given to determine
a law which the given series will obey.
p

210 COLLEGE ALGEBRA

The (m + l)th term is

%n = - H Hs 6 = 2 + 3 m (m — 1).

Hence u„^_^ = 3 (w — l)(w — 2) + 2.

Alsos„, = m2H ^ 4^^ ^b= 2m + m(m — l)(m — 2).

2. Find the mth term and the sum of m terms of the
series

1 + 4 + 11 4- 26 + 57 + 120 + ....

We have as before the series of differences

1 4 11 26 57 120
3 7 15 31 63
4 8 16 32
4 8 16

4 8
4

Here u^ — l^ ^u^ = 3, A^'w^^ = 4 for all values of r greater
than 1. (See note on previous problem.)

Hence

l^m ■ _

(:>H-(T)^+(:>-(;>+-

+ ( ^ U _ 3f ^M _ n'^ ] = 4 . 2'« - 3 - m = 2'»+2 _ 3 _ ^^^

SUMMATION OF SERIES 211

Also u^^;^ = 2"^+! - 2 - m.

+(s>+-+(:>-<:)-<T)-(2)=^"^^-^

263. If we are given but a few terms of a series without
being given the law of the series, we may determine a law
which the given terms will obey, but it is to be understood
that this may not be the only law which these terms obey,
for it may be possible to have the given terms obey different
laws. For example, the series

treated by the method of differences has for its nth term
(Sn^—9 7i-\- 8)a;"~i as is seen from Example 1 of 262. The
same series considered as a recurring series of order two has

2— 2ic
a generating function — ^ which yields as a gen-

T . JL — A X — jJ X

eral term

a — p

where a and fi are the reciprocals of the roots of

l-2a:-2a.^=0.

It is seen that the coefficient of x^ is a function of sym-
metric functions of a and ^8, and it can be proved that such a
function is rational, but it is more complicated than the

212 COLLEGE ALGEBRA

coefficient of the general term obtained by the method of
differences. It satisfies the recurrence formula

Hence the coefficient of x'^ by it would be 3^^ while from the
former standpoint it would be 3 • 5^ — 9 • 5 + 8 = 38.

Thus the two series agree to the fourth term but diverge
after this.

264. EXAMPLES

Find the mth term and the sum of m terms in the follow-
ing series:

1. 2-f-3 + 7 + 16 + 324----. 4. 3 + 8 + 10 + 10 + 9 + •••.

2. Y + 8H-11 + 17 + 31 + 62+.... 5. 2 + 5 + 10 + 9 + 6 + •.-.

3. _2 + 0 + l + 4 + 12+.... 6. 3 + 2-1 + 2 + 3-22 + ....

7. _4 + o + 18 + 5G + 120+--..

8. 2 + 5+9 + 15 + 25 + 43+--..

9. 4 + 7 + 12 + 21 + 38 + 71 + ....
10. 1 + 3 + 8 + 19 + 42 + 89 + ....

11. Find by the method of finite differences the (m + l)th
term of the series 2 + 5a; + 9:c2 + 15a;^ + 25a;* + 43a:;^+....

INTERPOLATION

265. If we have a series of terms, f(x), f(x + 1), /(a; + 2),
• . •,/(:r + n), we may put

f(x + 1) = Wj,
f(ix + 2) = u^,

fQx + 7l) = Wn-

SUMMATION OF SERIES 213

Whence, by 260,

f(x + n) = (1 + i^rfix) =/(a.) + Qa/(x)

This formula has been proven for the case where 7^ is a posi-
tive integer. The development given iov fQx + n) can, how-
ever, be proven to hold for all real values of n which make
the development convergent. It is often used as a formula
for interpolating values of a function between values corre-
sponding to integral values of x. Thus, given

/(I) = 2,
/(2) = 5,
/(3) = 10,

/(^) = 17,

to find /(l^). In this case x=l^ and we have

A/(l) = 3,
A/(2) = 5,
A/(3) = 7,
Ay(l) = 2,
A2/(2) = 2,
A3/(l) = 0,

A:/(1) = 0, r>2.

214 COLLEGE ALGEBRA

Therefore

/(1|) =/(l) + I A/(l) + J(f ^)a2/(1)

_ 9 _l_ 9 3_ _ 6 5 _ 4JL

"""^4 16~~16~^16*

As a verification we have the {n-\- l)th term,
f(n -H 1)=/(1)4- Qa/(1) + QaVCI)

when ?^ is a positive integer.

Put w + 1 = a:, or n = x — \^

then /(a:) = 2 + 3 (a: - 1) + (a^ - 1) (a: - 2) = 2;2 + 1,

which is true for more than two positive integral values of x
and therefore for all values of x. Whence

f(\^\ = 49 I 1 _ 65 — IJL
J \^i^) — 16 ^ ^ — 16 ^16*

As another example, given

log 50 = 1.698970,

log 51 =1.707570,

log 52 = 1.716003,

log 53 = 1.724276,
to find log 50.13.

Here x — 50, and we have,

log 50.13 = log 50 + (.13) A log 50 + (-^3)0^^-^) a2 log 50

(.13) (.13-1) (.13 -2) ^3 log 50.
3!

SUMMATION OF SERIES 215

This is as far as our data extend, since
A log 50 =.008600,

A log 51 = .008433,

A log 52 = .008273,

Anog 50 = -.000167,

A2 log 51 = -.000160,

A3 log 50 = .000007.
Hence
log 50.13 = 1.698970 + (.13) (.008600) + C-^^X-^^-l)

(-.000167) + (■1S)C1'^-^)C1^--) (.000007) = 1.700098.*

3 !

It may be noticed that the second term, (.13)Alog50, is
the amount added to log 50 by the usual interpolation to find
log 50.13. The succeeding terms furnish additional correc-
tions.

266. EXAMPLES

1. The cube roots of 60, 61, 62 are respectively 3.01587,
3.93650, 3.95789; find the cube root of 60.25.

The positions of a comet at Greenwich mean midnight are
as follows :

Eight Ascension Declination

h m s

March 15, 1907 6 48 27 - 12° 46'

March 19, 1907 6 40 10 - 9 28

March 23, 1907 6 33 18 - 6 26

March 27, 1907 6 27 44 - 3 42

* What function of a number a logarithm (log) is will be explained in the
next chapter.

216

COLLEGE ALGEBRA

What is

2. Its right ascension at Greenwich mean midnight on the
16th of March, 1907 ?

3. Its declination at Greenwich mean midnight on the 16th
of March, 1907?

A rifle was shot at different ranges and the following table
of elevations e for the vernier peep sight, for the given dis-
tances d was obtained.

d

e

0

21.0

100

24.5

200

28.5

300

33.5

400

40.0

500

48.5

4. What is the elevation for 425 yards ?

5. Solve generally and find what function of the distance
the elevation is.

CHAPTER XVI
LOGARITHMS

267. Definitions. If we consider the equation a^ = ?/, the
problem, given two of the three numbers, a, x^ ?/, to find the
tliird, leads to the consideration of the following types:

1. Given x and y^ to find a.

2. Given a and a;, to find y.

3. Given a and y^ to find x.

The solution of the first presents itself in the form a = v^,
by taking the 2:th root of both members of the equation.

The problem in the second is to raise a to the a;th power,

which operation is called exponentiation, In this operation

a is called the base and a^ the exponential of x with regard
to the base a.

In the third case the operation of finding x when a and y
are given is called the log arithmetic operation, and is ex-
pressed in symbols by the equation 2;=log«?/. From this
it is seen that the equation a^ = y may be written in the
form a^^'^'^y = y. The logarithm of a number y to the base " a "
is that exponent which indicates the power to tvhich the base
must be raised in order to produce y. The expression log^ y
is read ''logarithm of y with respect to the base a." It is
seen that log^?/, when found, is simply an exponent, and as
such is subject to the laws of indices.

268. Since a^ = 1 for all finite values of a different from
zero, it follows that log^ 1 = 0. Since a^ = a, it follows that
log^a = l.

217

218 COLLEGE ALGEBRA

Since a^"^ = go , and a""^ = 0, for all finite values of a > 1,
it follows for such values of a^ that log^ 0 = — oo and

log«QO = +QO.

Similarly for all positive values of a < 1, log„ 0 = oo and

log^QO =-0O.

If <^ > 0 and a; is a real number, a^ cannot be negative,
therefore the logarithm of a negative number is not real.

If a^ = m, a'^ = 7^, and m > n^ that is, if a^ > (X^, it is obvious
that x> y Avhen a > 1, that is, when the base is greater than
unity, the greater the number the greater the logarithm,
and conversely.

269. Theorems. Let a^ = w, a^ — n, then mn = a^a^ = a-^"^^.
Therefore, log^ (w>i) = x + y = log^ m + log„ w.
Thus, if logio 2 = 0.3010 and log^^ 3 = 0.4771,

then logio ^ = ^^^^lo ^ + logio 3 = 0.7781.

Similarly,

logaCmn •'•p')=loga(mn"-}-\-log„p

= l0gam -slogan -\- '" +l0gap,

that is, the logarithm of a product is equal to the sum of the
logarithins of its factors.

Ag-ain, — = — = a^~y.

^ n ay

Therefore, log^ { — \=x — y = log« m — log,, ?^,

that is, the logarithm of a quotient is equal to the logarithm of
the dividend minus the logarithm of the divisor.

Thus
logio 5 = If^gio 1^ - l^g'io- = 1.0000 - 0.3010 = 0.6970.

LOGARITHMS 219

Raising both sides of the equation a^ = m to the hih.
power, where k is any real number, integral or fractional,
positive or negative, we have a'^^' = m^.

Therefore, by definition, loga (m*) =k • x= k log« 771, that
is, the logarithm of any power of a nurnher is the logarithm of
the number 77mltiplied hy the index of the power ^ whether the
index he integral or fractional^ positive or negative.

Thus logio *^ = logio 2' = 3 log,, 2 = 0. 9080,

and log,„^2 = logio 2' = 1 log,„ 2 = 0.1008.

Note. Since the remainder, when divided by the divisor, gives a
quotient which is less than one half, it is neglected ; if that quotient were
greater than one half, it would be called unitij.

p
For a base a > 1, log^ ~ = logap — loga ^, which is positive

or negative according as p^qi that is, the logarithm of a
number greater than unity is positive and of a number less
than unity is negative.

If 1 > a > 0 and a^ > a^, then f - J > ( ) and therefore

y>x^ and the greater the number the less the logarithm, and

p
conversely. The log„ — = log„^ — log^^, which is positive

or negative according as p^q. Therefore with this base
the logarithm of a number greater than unity is negative
and of a number less then unity is positive.

We have seen, therefore, that for a base greater than unity
the logarithm of a number greater than unity is positive and of
a number less than uiiity is yiegative; ivhile for a base less than
unity the logarithm of a number greater than unity is negative
and of a number less than unity is positive. The same result

may be arrived at by substituting - for 5 in the transforma-
tion formula of 271.

220 COLLEGE ALGEBRA

270. EXAMPLES

1. If a series of numbers are in G. P. their logarithms are

in A. P.

2. Given 2-^ = 8, find x.

3. Given logg 27 = a;, find x.

4. Given log^ 32 = 5, find x.

Given log 2 =0.3010, log 3 = 0.4771, log 7 = 0.8451; find
the logarithms of the following numbers to the same base:

5. 14. 8. 32. 11. 14f 14. -^96.

6. 28. 9. 101. 12. 2.31. 15. v/48.

7. 24. 10. 20f 13. -^'M. 16. ^I375.

17. Find the logarithm of 243 to the base 9.
Let X be the required logarithm.

Then 9-^=243.

But 9 and 243 are both powers of 3. Hence 3^^ = 3^,
^^^ 2:r = 5, or 2;= 2.5.

18. Find the logarithm of 32 to the base 4.

19. Find the logarithm of 4 to the base 32.

20. Find the logarithm of 3^3 to the base 49.

21. Find the logarithm of 343 to the base ^^.

22. Find the logarithm of ^V to the base |.

271. Let log^ n = x^ log^ n = y^ then

a'' = 71 and h^ = n
and therefore 0^ = 1^.

LOGARITHMS 221

Taking the logarithm of both members of the last equation
with respect to any third base c, we have

or by substituting the values of x and ?/,
log« '^ log, a = logft n log, b,

whence log^ n = — ^^^ log^ n.

Since o was any real number whatevei', the ratio — ^^^^— is

constant for any given value of a and ^, and is called the
moduhis of the tra7isformation.

T^ 1 loo", a 1

i^or c= a, we liave ^'^ —

log, 5 log,,^'

and log5 n = - — — log„ n.

loga 0

This is a formula for transforming logarithms of numbers
which are known with respect to a base a into logarithms of
those numbers with respect to any other base b.

The student may show that log,^ b log^ a = l.

272. Although theoretically any positive number except
unity could be made the base of a S3^stem of logarithms, yet
for practical purposes only two systems are at all frequently
used. One, called the natural system^ or sometimes the
Napierian system^ is explained in a subsequent article, 283.
The other, known as the Briggs^ or common^ system^ has the
number 10 for its base. It is used for all purposes involving
merely numerical calculation. The advantage of this base
consists in the fact that any change in the position of the
decimal j)oint in a number will merely add an integer to, or
subtract it from, the logarithm, because the number will then

222 COLLEGE ALGEBRA

merely be multiplied or divided by an integral power of 10,
269. Hence the fractional portion of the logarithm will be
the same for any sequence of figures, whatever the position
of the decimal point among them.

273. In the Briggs system the logarithm of 1 is 0, and the
logarithm of 10 is 1, by 268. Hence the logarithm of any
number between 1 and 10 is a positive fraction. Every
number greater than 10 may be obtained by multiplying a
number between 1 and 10 by an integral power of 10. Hence
its logarithm consists of an integer plus a fraction. This
integer, which has been shown to be dependent merely upon
the position of the decimal point in the number, is called the
characteristic of the logarithm. The fractional part, which
depends merely upon the sequence of figures in the number,
and is ordinarily written in the form of a decimal, is called
the mantissa of the loga7'ithn. The mantissa is ordinarily not a
terminating decimal, but is carried out four, five, six, etc. places
according to the degree of accuracy required in the work.

Thus log 2 = 0.3010, and log 200 = 2.3010. In each case
the mantissa is .3010, while the characteristics are, respec-
tively, 0 and 2.

274. The logarithm of a positive number less than unity
is really negative, but since such a number may be derived
from a number between 1 and 10 by dividing it by an in-
tegral power of 10, it is convenient to regard the logarithm
as composed of a positive mantissa and a negative charac-
teristic. Thus if log 2 = 0.3010, then since .002 = -?-,
log .002 = 0.3010-3.

275. Two methods are in common use for writing loga-
rithms with negative characteristics. Thus log .002 = 3.3010,
where the negative sign is placed over the characteristic to
indicate that it alone is negative ; or the — 3 is called 7 — 10

LOGARITHMS 223

and tlie logarithm is written 7.3010 — 10. By this means
the negative portion is always a multiple of 10, and is kept
quite separate from the positive portion of the logarithm.

276. To find the logarithm of a given number.

1. To find the characteristic. In 273 we have seen that if
the decimal point follows the first significant digit, the charac-
teristic of the logarithm is 0. For every place the decimal
point in the number is moved toward the right the number is
multiplied by 10. Hence if the number is greater than 10,
the characteristic is one less than the number of significant
figures to the left of the decimal point. Likewise for every
place the decimal point is moved toward the left the number
is divided by 10. Hence if the decimal point immediately
precedes the first significant digit, the characteristic of the
logarithm is — 1, or 9 — 10. If one cipher intervenes, it is
8 — 10, and so on, subtracting one for each additional cipher.

The characteristic should be written first, and always ex-
pressed even though it be zero, in order to avoid error due to
forgetting it.

2. To find the mantissa from the table, (a) When the
number has just three significant figures. Pages 224 and 225
give a table of the mantissoe of the Briggs logarithms of all
integers from 1 to 1000. In order to find the mantissa of a
given number, look for the first two digits in the column
marked N. These indicate the row in Avhich the mantissa
is to be found. The column is designated by the third digit.
Thus the mantissa for 478 is 6794, and the entire logarithm
is 2.6794 by 1.

(5) When the number has less than three significant
dibits. To find the log^arithm of .7 we look for 700 in the
tables and find the mantissa 8451. Hence log .7 = 1.8451,
or 9.8451-10.

224

COLLEGE ALGEBRA

N

0

1

2

3

4 5

6

7

8

9

0

0000

0000

3010

4771

6021

6990

7782

8451

9031

9542

1

0000

0414

0792

1139

1461

1761

2041

2304

2553

2788

2

3010

3222

3424

3617

3802

3979

4150

4314

4472

4624

3

4771

4914

5051

5185

5315

5441

5563

5682

5798

5911

4

6021

6128

6232

6335

6435

6532

6628

6721

6812

6902

5

6990

7076

7160

7243

7324

7404

7482

7559

7634

7709

6

7782

7853

7924

7993

8062

8129

8195

8261

8325

8388

7

8451

85i3

8573

8633

8692

8751

8808

8865

8921

8976

8

9031

9085

9138

9191

9243

9294

9345

9395

9445

9494

9
10

9542

9590

9638

9685

9731

9777

9823

9868

9912

9956

0000

0043

0086

0128

0170

0212

0253

0294

0334

0374

11

0414

0453

0492

0531

0569

0607

0645

0682

0719

0755

12

0792

0828

0864

0899

0934

0969

1004

1038

1072

1106

13

1139

1173

1206

1239

1271

1303

1335

1367

1399

1430

14

1461

1492

1523

1553

1584

1614

1644

1673

1703

1732

15

1761

1790

1818

1847

1875

1903

1931

1959

1987

2014

16

2041

20(J8

2095

2122

2148

2175

2201

2227

2253

2279

17

2304

2330

2355

2380

2405

2430

2455

2480

2504

2529

18

2553

2577

2601

2625

2648

2672

2695

2718

2742

2765

19
20

2788

2810

2833

2856

2878

2900

2923

2945

2967

2989

3010

3032

3064

3075

3096

3138

3139

3160

3181

3201

21

3222

3243

3263

3284

3304

3324

3345

3365

3385

3404

22

3424

3444

3464

3483

3502

3522

3541

3560

3579

3598

23

3617

3636

3655

3674

3692

3711

3729

3747

3766

3784

24

3802

3820

3838

3856

3874

3892

3909

3927

3945

3962

25

3979

3997

4014

4031

4048

4065

4082

4099

4116

4133

26

4150

4166

4183

4200

4216

4232

4249

4265

4281

4298

27

4314

4330

4346

4362

4378

4393

4409

4425

4440

4456

28

4472

4487

4502

4518

4533

4548

4564

4579

4594

4()09

29
30

4624

4639

4654

4669

4683

4698

4713

4728

4742

4757

4771

4786

4800

4814

4829

4843

4857

4871

4886

4900

31

4914

4928

4942

4955

4969

4983

4997

5011

5024

5038

32

5051

5065

5079

5092

5105

5119

5132

5145

5159

5172

33

5185

5198

5211

5224

5237

5250

5263

5276

5289

5302

34

5315

5328

5340

5353

5366

5378

5391

5403

5416

5428

35

5441

5453

5465

5478

5490

5502

5514

5527

5539

5551

36

5563

5575

5587

5599

5611

5623

5635

5647

5658

5670

37

5682

5694

5705

5717

5729

5740

5752

5763

5775

5786

38

5798

5809

5821

5832

5843

5855

5866

5877

5888

5899

39
40

5911

5922

5933

5944

5955

5966

5977

5988

5999

6010

6021

6031

6042

6053

6064

6075

6085

6096

6107

()117

41

6128

6138

6149

6160

6170

6180

6191

6201

6212

6222

42

6232

6243

6253

6263

6274

6284

6294

6304

6314

6325

43

6335

6345

6355

6365

6375

6385

6395

6405

6415

6425

44

6435

6444

6454

6464

6474

6484

6493

6503

6513

6522

45

6532

6542

6551

6561

6571

6580

6590

6599

6609

6618

46

6628

6(i37

664(5

()()56

6665

6675

6684

6693

6702

6712

47

6721

6730

6739

6749

6758

6767

6776

6785

671H

6803

48

6812

6821

6830

6839

6848

6857

68()6

6875

6884

(;S!)3

49

6^X)2

6911

6920

6928

6937

6946

6955

6964

6972

6981

N

0

1

2

3

4

5

6

7

8

9

LOGAKITIIMS

225

N

0

1

2

3

4

5

6

7

8

9

60

6990

6998

7007

7016

7024

7033

7042

7050

7059

7067

51

7076

7084

7093

7101

7110

7118

7126

7135

7143

7152

52

7160

7168

7177

7185

7193

7202

7210

7218

7226

7235

53

7243

7251

7259

7267

7275

7284

7292

7300

7:308

7316

54

7324

7332

7340

7348

7356

7364

7372

7380

7388

7396

55

7404

7412

7419

7427

7435

7443

7451

7459

7466

7474

5()

7482

74W

7497

7505

7513

7520

7528

15m

7543

7551

57

75.")!)

7566

7574

7582

7589

7597

7604

7612

7619

7(527

58

7634

7(542

7649

7657

7664

7672

7679

7686

7694

7701

59
60

7709

7716

7723

7731

7738

7745

7752

7760

7767

7774

7782

7789

779(5

7803

7810

7818

7825

7832

7839

7846

(31

7853

7860

7868

7875

7882

7889

7896

7903

7910

7917

62

7924

7931

7938

7945

7952

7959

7J366

7973

7980

7987

03

7993

8000

8007

8014

8021

8028

8035

8041

8048

8055

64

8062

8069

8075

8082

8089

8096

8102

8109

8116

8122

65

8129

8136

8142

8149

8156

8162

8169

8176

8182

8189

66

8195

8202

8209

8215

8222

8228

8235

8241

8248

8254

67

8261

8267

8274

8280

8287

8293

8299

8;306

8312

8319

68

8325

8331

8338

8344

8351

8357

8363

8370

8376

8382

69
70

8388

8395

8401

8407

8414

8420

8426

8432

8439

8445

8451

8457

8463

8470

8476

8482

8488

8494

8500

8506

71

8513

8519

8525

8531

8537

8543

8549

8555

8561

8567

72

8573

8579

8585

8591

8597

8603

8609

8615

8(321

8627

73

8633

8639

8645

8651

8657

8663

8(5(39

8675

8681

8(38(5

74

8692

8698

8704

8710

8716

8722

8727

8733

8739

8745

75

8751

8756

8762

8768

8774

8779

8785

8791

8797

8802

76

8808

8814

8820

8825

8831

8837

8842

8848

8854

8859

77

8865

8871

8876

8882

8887

8893

8899

8904

8910

8915

78

8921

8927

8932

8938

8943

8949

8954

8960

8965

8971

79
80

8976

8982

8987

8993

8998

9004

9009

9015

9020

9025

9031

9036

9042

9047

9053

9058

9063

9069

9074

9079

81

9085

9090

9096

9101

9106

9112

9117

9122

9128

9133

82

9138

9143

9149

9154

9159

9165

9170

9175

9180

9186

83

9191

9196

9201

9206

9212

9217

9222

9227

9232

9238

84

9243

9248

9253

9258

9263

9269

9274

9279

9284

9289

85

9294

9299

9304

9309

9315

9320

9325

9330

9335

9340

86

9345

9350

9355

9360

9365

9370

9375

9380

9385

93f)0

87

9395

9400

9405

9410

9415

9420

9425

94130

9435

f)440

88

9445

9450

9455

94(50

9465

9469

9474

t|479

9484

9489

89
90

9494

9499

9504

9509

9513

9518

9523

9528

9533

9538

9542

9547

9552

9557

9562

9566

9571

9576

9581

9586

91

9590

9595

9600

i)(505

9609

9614

9619

9624

9628

9633

92

9638

9(543

9(547

m52

9657

9661

9666

mn

9675

9680

93

9685

9(589

9694

9699

9703

9708

9713

9717

9722

9727

94

9731

9736

9741

9745

9750

9754

9759

9763

9768

9773

95

9777

9782

9786

9791

9795

9800

9805

9809

9814

9818

96

9823

9827

9832

9836

9841

9M5

9850

9854

9859

9863

97

98(i8

9872

9877

9881

9886

9890

9894

9899

9<X)3

9908

98

9912

9917

9921

9926

9930

99:i4

9939

9943

9948

9952

99

9956

9961

9965

99(59

9974

9978

9983

9987

9991

9996

N

0

1

2

3

4

5

6

7

8

9

9f.

226 COLLEGE ALGEBRA

(6?) When the number has more than three significant
digits.

Thus in the case of log 32.456, since 32.456 lies .5Q of the
way from 32.4 to 32.5, its logarithm must lie about .56 of
the way from log 32.4 to log 32.5. But log 32.4 = 1.5105
and log 32.5 = 1.5119. Hence, leaving the decimal point
out of account, the increase, or tabular difference as it is
called, is 14, and .36 of this is 7.84. Hence, adding this cor-
rection to log 32.4, we have log 32.456 = 1.5113. Since our
tables are given only to four decimal places, we retain only
four in correction, always figuring the fourth place to the
nearest unit, thus in this case adding 8 as the correction.

Likewise for log .0035678, we find from the tables log
.00356 = 7.5514 - 10, and the tabular difference is 13. The
correction (13 x .78) is 10, so that log .0035678 = 7.5524-10.

Find the logarithms of the following numbers :

1. 428. 5. .524. 9. .050009.

2. 327. 6. .02345. 10. .0042085.

3. 82.46. 7. .38634. ii. 20.308.

4. 32.875. 8. 430.23. 12. 7,352,000.

277. To find the number corresponding to a given logarithm.

Given log x= 3.3765 ; to find x.

This is the reverse operation of that given in 276. Since
the characteristic merely determines the position of the
decimal point, 273, we look for the mantissa in the table.
The next smaller mantissa in the table is 3747, which corre-
sponds to the number 237. The excess of 3765 over 3747 is
18. The tabular difference as found from the table by sub-
tracting 3747 from 3766 is 19. Hence 3765 is If of the way
from 3747 to 3766, and the corresponding number is about
i| of the way from 237 to 238, or, reducing IJ to a decimal.

LOGARITHMS 227

about .95 of a unit beyond 237. Hence the corresponding
digits are 23795, and since the characteristic is 3, the deci-
mal point must be moved three places to the right of where
it would be placed for a zero characteristic, and hence
a; = 2379.5.

Similarly, if log ^ = 7.3765-10, ?/ = . 0023795, since in
this case the characteristic is 7 — 10, or — 3, and the decimal
point must be moved three places to the left.

Find the numbers corresponding to the following loga-
rithms :

1.

2.8987.

7.

4.6062.

13.

5.8124.

2.

3.5705.

8.

2.6842.

14.

0.7318.

3.

0.7016.

9.

1.3427.

15.

4.7306-10.

4.

9.8814-

-10.

10.

0.4850.

16.

5.4783.

5.

1.9542.

11.

7.6123.

17.

2.7005.

6.

3.6558.

12.

8.5493-

-10.

18.

1.6100.

278. Cologarithms. The cologarithm of a number is the
logarithm of the reciprocal of the number.

Thus colog 425 = log -^^ = log 1 - log 425 (269)

425

= 0 - 2.6284.

But since we always wish to have the mantissa of a loga-
rithm positive, we write 0=10—10, and subtract 2.6284
from this, as follows:

log 1 = 10.0000 - 10
log 425= 2.6284

colog 425= 7.3716-10.

228 COLLEGE ALGEBRA

In practice this is done mentally by beginning at the left
and subtracting each digit from 9, except the last significant
digit, which is subtracted from 10.

279. Computation by logarithms.

EXAMPLES

1. Find the value of 2345 x 2.327 x .004296.

log (2345 X 2.327 x .004296) = log 2345 + log 2.327

+ log .004296. (269)
log 2345 = 3.3702
log 2.327 = 0.3668

log.004296 = 7.6331 -10

log (2345 X 2.327 x .004296) = 1.3701. '

3692
19)9.00
.47
Therefore, 2345 x 2.327 x .004296 = 23.447.

Note. It must be borne in mind that these logarithms are only approxi-
mations carried out to a number of decimal places determined by the number
of decimal places of the table used. Four-place tables do not give even five
figures in the result with any considerable degree of accuracy. If greater
accuracy is desired, a table with more decimal places must be used.

2. Find the value of '-^ '—^ — -•

.002578 X 386.5

Since this is the product of two numbers divided by the
product of two others, it might be solved by subtracting the
sum of the logarithms of the factors of the denominator from
the sum of the logarithms of the factors of the numerator.
But this would require several operations, and it is customary
to regard the fraction as the continued product of the factors
of the numerator and the reciprocals of the factors of the
denominator.

LOGARITHMS 229

Thus

37.54 X .02486 ^ /g^^^^ ^ ^^2436 x -^ x -^.^
• ^ .002578 X 886.5 "^ V .002578 386.5;

= log 37.54 + log .02436 + colog .002578

+ colog 386.5.
log 37.54 = 1.5745

log .02486=8.3867-10

colog .002578 = 2.5887

colog 386.5 = 7.4128 -10

^87.54 X. 02486 ^^g^,^_^Q^
^ .002578 X 386.5

Hence 87.54 x .02436 ^ ,^,,3^

.002578 X 386.5

3. Find the value of (38.64)6.

log (38.64)6 = 6 X log 38.64. (269)

log 38.64 = 1.5870.
Therefore log (38. 64)6 ^ 9. 5220,

and (38.64)6 = 3,326,900,000.

4. Find the value of V7f684.

log V.7684 = ilog .7684.
log .7684 = 9.8856-10
= 49.8856-50.
Therefore log </7mi = 9.9771 - 10,

J'

5j

and V. 7684 = .9486

230 COLLEGE ALGEBRA

Find by means of logarithms the approximate values of
the following expressions :

6. 234 X 345 X 456. ^ 48. 8T x .03245 x 389.2

6. 2345 X 3456 x 4567.

10.

7.459 X 9.351
.07371 X 907.3 x .6007

7. 86.23 X 23.45 X. 08632. ^^' .7623x8.076

28.42 X 67.54x96.72 ^^ -7.831x3.867x8.903
92.57 X 13.83 * " .007459 x 12.87

Note. Since all real powers of 10 are positive, negative numbers have
no real logarithms. Hence examples involving negative numbers must be
worked as if the numbers were positive, and then the proper sign is to be
attached to result.

^2 r-. 008734) X (- 8.345) x 834.7
(-.8793) X (-900.6)

13. (8.341)*. 19. (-.5429)1

''' ^''''''^'' 20. (-.05387)1.

''' (^^•^^);- 21. (.3841)1

16. (56.38)^. ^^ V3596 x ^:4287

17. (.02583)i a/. 0586

18. (-8.425)1 23. aJ-V:5804 x V^MO^.

24. Find by means of logarithms the amount of f 486 in
five years at five per cent if the interest is compounded
annually.

25. Find the amount of -1384 in forty years at four per
cent if the interest is compounded semiannually.

26. Solve the equation 4^ = 246.

LOGARITHMS 231

Taking the logarithm of each member, we have

X log 4 = log 246,

lo^ 246 2.3909 . o-i

or x = -^ ■ = T = 3.9<1.

log 4 0.6021

27. Solve the equation 415^^^^ = SIT'''^"^

28. Find the logarithm of 428 to the base .8.

If X represent the required logarithm,

.8^^=428,

, los^ 428 ^^ .,1 otri\

and X = — (Compare with 271)

log .8

2.6314 2.6314 ^r. -.^

= — z< .lb.

9.9031-10 0.0969

Note. In this case 9.9031 — 10 is really a binomial expression and hence
must be combined in a single monomial before ordinary arithmetical divi-
sion can be performed.

29. Find the logarithm of 376 to the base 12.

30. Find the logarithm of .536 to the base 7.

thp: exponential function

280. Definition. Let us deiine F(x^ as the limit of ll -\ — ) ,
as w = CO, that is, -F(a:) = L f 1 + - j , for both real and complex

7 /• 7 71=30 V nJ

values oj x ana n.

281. From the foregoing definition we easily derive the
following properties of FQx~).

1. When x=0, we have ^(0) = L(iy = 1.*

* Although I*' must be considered an indeterminate form when it is the
limit of a variable which approaches 1 raised to an infinite power, here we
have strictly a constant.

232 COLLEGE ALGEBRA

2. For all real values of x and n^ F(x) is the same for a
given value of x in whatever way n becomes indefinitely
great. For if n is any real positive fraction, we can always
choose two consecutive positive integers such that

Hence, according as x is positive or negative,

X <x<:^ X

m

+ 1 -^ 'nr m

and l4.^>l + ^>l+_^,

and hence (l + -T^^ > f 1 + - Y' >(l +
\ mj \ nj \

1 +

m + 1,

V m + iy

Taking the limits as n = oo, and hence as w = go, and
771 -j- 1 = 00, and denoting by /(a;) the value of the limit of

(l-\ — ) as m, while remaining a positive integer, becomes
\ 'rnj , ^

indefinitely great, we see that ( 1 + - j , lying between two

numbers each of which approaches the limit /(a:;), must itself
approach /(rr) as its limit, or

when n is positive, whether x is positive or negative.

* For x < 0, the inequality sign and the order of the exponents would be
reversed.

LOGARITHMS 233

li n = — p^ where jt? is a positive number, then

n=oo\ ^/ p=ao\ pJ p^coXp X.

p-x^^\ p — xj \ p — Xj

We have proved therefore that
for all real values of x and n.

If :?: = 1, we have ifl + iY = /(!).

3. Since X fl + -Y' = l{[\^- ^ p Y ,
we have, putting - = m,

or f(x) = /(1)% for all real values of x.
In particular, /( —a;) =/(l)"^.

4. By (3), we have

or /(^)/(i/)=/(^ + i/).

The second property shows that for all real values of x
^'''^^' Fix)=f(ix^.

234 COLLEGE ALGEBRA

282. Expanding f 1 -f - J by the binomial theorem, we have

1 + - =l+-x+ ^ — + •-

\ nJ n z I n^

1-1

+ -^ -. --+ -" =1+2; + — —, — x^ +

r\ 7f ^ I

i_iYi_2v.Yi_!i^

+ ^ ^^ + •••• (1)

Taking the limits of both sides as w = go, we have

That the right-hand member has this limit may be seen as
follows :

The (r + l)th term is

i_iYi-2V..ri-':^

. y^ n/\ 71/ \ 71 y r
tr+i = X .

r I
Let fl^j, «2» ^3' '"•) ^^6 positive proper fractions. Then, since

(1 — «i)(l — ^2) = 1 — <^i — «2 + ^1^2'

1 > (1 — «i) (1 — ^2) > 1 — «i — «2-
Similarly

1 > (1 — <^i)(l — (^>2)0^ ~ ^3) > 1 — ^1 — «2 ~~ ^3'

l>(l-a{)(l-a^) ••• (1--%.)>1 -aj-«2-^3 "• - ^k-

(2)

LOGARITHMS 235

1 2 r — 1

Choosinsc A: = r — 1, a, = -, «„ = -, •••, cik = -> ^^^ multi-

r n n n

plying (2) by — -, we have, using a positive x for convenience,

^1

x^ \ti n n

^ <" f > ^ ->

l_,l + 2^,..+i^

^^ ^\ > ^r+l >Z1~

x^ , x"" J rCr — 1) ^

r ! ^^^ r\ n r\

X

X . x^ x^

-r>^r+i>-r-

r\ '"-^ r\ 2n (r-2)!

Giving r the values 0, 2, 4, •••, and the values 1, 3, 5, •••,
we get the following two systems of relations respectively,

JL — Ci — X, X — Cq — •*/,

/yi^ /yi^ /yt^ /y^ /yn> nriO

2!^ ^^2!~27i' 3T^ ^^3T~2^' ^^

Adding the corresponding terms of the relations of each
system, we have

x^ . X^ , . , . . . ^ . ^ -I . x^ , x^

2! 4!

> ^1 + ^3 + ^5+ •••>^ + 2: + 47+ -

2 TiV 214!
-2^1" +3! + 61+-

236 COLLEGE ALGEBRA

As 7i = oo, the series 1 + ^ + — 4- •••

and ^ + 3T + It +

are absolutely convergent, 212, 16, the terms

27iV 2! 4! J

and f^f^+^ + ^ +

2 7iV 3! 5!

approach zero, and hence we see that

W=00 O i O I

Adding and subtracting these limits and remembering
that

^l + ^2 + - + ^n+l= I4-- ,

X

^l-^2+- + (-lr^«+l= 1-- h

n.

X

n

71

wehave l (1 + ^^ =1 +. + f, + ^^ + ... , (213}

n^^\nj 213!

xv ^ . x^ x^

^ 7iJ 2! 3!

LOGARITHMS . 237

This completes the proof that for any real values of x and n^

xY -, . , x^ . x^

If in the preceding proof we had taken x negative, the
right-hand column of relations (3) would have had their
signs all reversed, but in taking the limits we would have
arrived at the same result.*

283. The series

is denoted by e, which is the base of the natural system.

That the quantity e is finite and lies between 2 and 3 may
be easily seen, for

6=1 + 1 + 1 +

_ 2+

11 11

and since ^ '^ Y\'^ T\^ '" ^^ ^ 2^ ^^'^ '**'

, 1

or e — 1<

^ 2

or e — 1 < 2.

Therefore ^ < 3, hence 3 > e > 2.

Since /(a:) =/(l)-^, we have proved that

Zt I 6 1
for all real values of x.

-n ) -1

* Or Otherwise ^ fi-^V^ L \fi^_±_\ "1

fix) jo^y

238 • COLLEGE ALGEBRA

284. The determination of the value of F(x) when x and
n are complex numbers can be made as follows :

If 2 is a complex number and n a positive integer, the (r + 1) th

term in f 1 + - j ^ z being equal to a; (cos (j) + i sin ^), is, 380,

T =

[l jfl — -j---fl ja:^(cos r(l)-{-i sin r(f))

1 )(1 — -j--.(l ]x^ cos rcf)

Liet t ^+j — —^ ,

1 _ IVl _ ?y . Yl _ r^Vr sin r<^
nj\ nJ \ n J

and ^'',.+1 =

'•+1 — ^ J

then as before 1 = ^'.^ = 1,

X cos (^ = t' ^ = X cos <^,

-cos2^>^'3>— cos2(/, ^— ^,

^^ o / 4/ ^^ o ji ^ cos 3 (f)
— - cos 3 (/) > ^'4 > — cos 3 (/) ^,

X^

5

, , ., a;* 1 JL ^* cos 4 6
-cos5(^>t'6>— cos5(^-^— ^,

x^ ± ^ .1 -^ x^ , x"^ COS n 6

nl 71 ! 2 n (n — 2) !

LOGAIUTHMS 239

On adding, we get

l+a;cos<^ + — -cos 2<j)-\ \-—-cos7i(t)>t'.-\-t'-\ h^Wi

x^
> 1 H- a; cos <^ + • • • H r cos n^

n\

x^ f x^~^ \

cos 2 6 + X cos 3 (^ + • • • H —— cos ncj) ) .

2n\ (»i — 2) ! /

As n = 00, the series

1 -\-x cos 6 + • • • H r cos n<j>

n !

and cos 2^ -\- x cos 3 <^ H- • • • H ——cos w<^

are absolutely convergent, since the terms of each are less
than the corresponding terms of the absolutely convergent
series o „

z ! n\

which as 7^ = GO has the limit /(a:) or e^. Therefore

2

L (^\ + ^'2+'-' + ^^+i) = l + -^'co3<^ + |-cos2<^+.... (1)

In the same way it can be proved that
L (^''2 + ^''3+-+^Vi)=^sin</> + ^sin2(^+--. (2)

Multiplying (2) by i and adding the result to (1) and observ-
ing that

V 1 ^^=^ -^ 1 ' ^9 "t" ^ ^ 2 ~~ 9' ^ "I 3 ~~ 3' etc.,

and that ( T^ + T^ + • • • + ^+1) = (l + |T , (213, 214)

240 COLLEGE ALGEBRA

we liave

L (T,-hT^+'- + T^^O = L fl + -Y = l + 2;(cos(/, + zsin(^)

71=00 n=^au \ fl/

+ — (cos2(^ + ^sin2(/)) H ,

The extension to any real value of n may be made by
considering the limits of the moduli and arguments of

1 -\ ,1+-,1h — and finally extending to

m + ly \ nj \ mj

a negative n.

Hence, when z is complex and n is real,

If both z and n are complex, we have, if
n = m (cos (f) -{-i sin 0),

X 1 + ^1 =£ (1+ ^

7i/ 7n=oo v m (cos (^ + *' sin (^^

— L f(l-\- ^(CQS (^ - ^ sin (f)~^yn\ cos <t,+i sin ^
m=x> W m J J

= (/(2(cos(^-2sin<^)))^««'^ + ^'«^"*^.

This is as far as we can carry the proof. If, however, we
agree to give to a complex exponent such an interpretation
that the third property, viz. f(z) = (/(l))^, shall still hold
even when z is complex, we have

F{Z^ = (/(^(COS (^ - Z sin <^)))cos« + ism<^

LOGARITHiMS 241

Thus, for all values of x and 71, we have

Fix) =f{x).

That the series denoted by f{x) is convergent has been
seen from the mode of its derivation, since each of the con-
stituent series of which it is composed is convergent whether
X be real or complex. The result

1 + 2) =,- = !+:,+ £- + |.^ + ...

is known as the Exponential Theorem,
285. Since a''= e^^o^e", we have

a-=l +(log,a) a;+ (log,a)2|j + -. (1)

Substituting in this 1 + y for a, we have

(l+2,)- = l + a;logXl+y) + 5(log,(l+y))2+.... (2)

If y is numerically less than unity, we can expand by the
binomial theorem and get

-, , , x(x—X) 2 , x(x — X)(x — ^^ q ,

i + ^^ + -^-2| — -y -^— ^7 -y^-^'"

= \^x log/l + y) + ^ (log,(l + y))2 + .... (3)

Equating the coefficients of x on both sides of (3), we
have (212, 17)

log,(l + ^)=y-f + f-^^V-. (4)

This is called the logarithmic series.
Changing y into — ^, we have

log,(l-^)= -^-|--|--^ . (5)

242 COLLEGE ALGEBRA

The logarithmic series may be used to find the logarithm
of any number, but since the series converges so slowly, it is
more expedient to use the following:

Subtracting the corresponding members of (4) and (5),
we have

ioge(i + y') - ioge(i -y^=\y~\'^\ — )

\ -^ 2 3
log,J-±^=2(y + | + ^+...). (6)

^ T I 7/ /yyi fvyj 'Y)

Substitute in this ^ = — , that is, y = , and it becomes

1 — y n m-\-n

^ rn^^jm-n\^m-n\^^ljm-n\^^ ...Y (7)
n \m 4- n 3 \m + 7iJ 5 \m + nj •'

If m = n^ logg 1 = 0.

If. = 2a„d« = l,log.2 = 2g + lg)Vlg)%.

If m = n + 1, (7) becomes

log. '±^ = 2{,,^^ + i(,r^^]+iJ,r^^] +

n "\2n+l iy2,n + lj 5V2» + ]

and this is equivalent to

1^ 1 ^V-l- (8)

5\2n-\- 1

LOGARITHMS 243

From (8) the logarithms of all numbers to the base e may
be obtained. It has been shown in 271 how to change from
one base to another.

To obtain the logarithms of numbers to the base 10 we
have to multiply the logarithms of the numbers to the base e

by , which may be found from (8) to be 0.434294+ .

log, 10

Examples. Compute log^ 2, log, 3, loge4, logjQ2, logio3»

CHAPTER XVII

DETERMINANTS

286. If in algebra certain forms are of frequent occurrence,
it is convenient to have a suitable notation to express tliem;
thus the expressions, a^^ — aj)-^^ ^i^2^3 ~l~ ^j^2pi + ^3^i^2 ~ ^3^2^!
— a^^c^ — a^^c^^ are instances of such forms which often
occur. The first may arise as the result of eliminating x and

y from the equations

a^x + a<2^y = 0,

h^x 4- h^^y = 0.

The second from eliminating x^ ?/, z from the equations

a^x 4- ci^y + ^3^ ~ ^'
h^x + h^y + h^z = 0,
c^x + c^y 4- c^z = 0.

In general, if we eliminate the n variables from a system

of n linear homogeneous equations, we get a result of the

form _ , 7 /x

z ± a-^o^c^ ■•'I,, = 0.

These forms are called determinants^ and since they are
functions of the coefficients only and may be considered
without reference to their origin, the preceding being but
one way in which they arise, the definition of a determinant
should obviously contain no reference either to its origin or
to the variables.

244

DETERMTXANTS

245

287. Definition. If tve have ^t^ quantities arranged in a
square of n roivs and n columns^ then the sum^ with j^roper signs
of all the terms that can he formed by taking the product of n
quantities one and ojily one from each row and one and only
one from each column^ is called the determinant of those quanti-
ties and is said to be of the nth order.

The sign factor for any term of the determinant as defined
above is determined by writing in succession the numbers of
the rows from which the quantities composing it have come;
and in a separate series the numbers of the columns and tak-
ing + or — according as the total number of inversions* of
order in the two series is even or odd. Since the factors of
any term may be written in any order whatever, we may
obviously write them so that the numbers in one of these two
series are in the natural order (order of magnitude) and
therefore in determining the sign factor of a term we need
only take account of the inversions in the other series.f

288. Notations and Definitions. The ordinary notation
for a determinant is

1 ^*>

^1 h

for «j^2 ~ ^2^1 '

a^ a^

^3

h K

^3

^1 ^2

^3

iova^K^c^+aJ)^c^ +

«1 «2 ■

^1 ^2 ••

'l h ■

■■ i„

for S ± a^^c^'-'l^.

* Whenever a greater integer precedes a less there is said to be an inver-
sion of order.

t Show that the same sign factor results whatever be the order of the
series, pairs being kept together.

246 COLLEGE ALGEBRA

The quantities a^ a^, etc., are called the constituents or ele-
ments^ and the products a-J)^, (t-^b^c^, etc., are called the terms
of the determinant.

In the square array representing a determinant the diag-
onal from the left-hand top corner to the right-hand bottom
corner is called the principal diagonal; and the diagonal
from the left-hand bottom corner to the right-hand top
corner is called the secondary diagonal. The term formed
by the product of all the constituents along the principal
diagonal is called the principal or leading term.

Another convenient notation for determinants is to write
each constituent with a double sufiix, the first indicating the
row and the second the column to which the constituent
belongs. Thus the determinant of the ni\\ order in this
notation is written

^21 ^^22 ' ' * ^2't

, or simply (a^^ a^^ ••• a„„).

Elements are said to be co7ijugate to each other when the
place that either occupies in the row is the same as the
other occupies in the column. Thus ajj. and aj^^ are conju-
gate elements. Elements along the principal diagonal are
self- conjugate.

289. The principal term is a-^-^ a^^ a^^ a^^ '"(^nn^ and we
can get all the other terms from this by interchanging the
second (or the first) series of suffixes in all possible ways,
for this gives all possible ways of taking one and only one
element from each row and column. If we take any term
a^i^ ^2,- % ••• a„i^ where the fs form a permutation of the
numbers 1, 2, 3, 4, ••• n^ and interchange two adjacent fs,

DETERMINANTS

247

we obtain another term of the determinant which has a sign
opposite to that of the given term, for we have either one
more or one less inversion among the is. If any two fs
having r suffixes between them are interchanged, the result-
ing term will have a sign opposite to the given term, for the
interchange can evidently be brought about by 2 r -h 1 suc-
cessive interchanges.

EXAMPLES

1. Determine the sign of ^j^ ^22 ^33

a^^ considered as a

term of

a

a

11

31

^41

12
^22
'32
i^42

a

a

a

13

33
43

a

The constitutents are already written down in the order
of their rows. The series of numbers denoting the columns
from which the constituents have come is 1, 2, 3, 4, and
contains no inversions of order. The sign of the term is
therefore +.

2. For the same determinant find the sign of a^^ a^i a^2 ^w
To determine the sign in this case we have to determine
the number of inversions of order in the series 3, 1, 2, 4.
As there are two inversions of order, viz. 3 before 1 and 3
before 2, the sign is + .

3. For the same determinant determine the signs of tlie

following terms : a^^ a^i a^^ ^43 ; <^i2 ^21 ^

34 ^43 ' ^

13 ^^24 '^31 ^

42*

4. For the determinant

^1

«2

h

h

^1

^2

d.

cL

a.

do d.

248 COLLEGE ALGEBRA

determine the sign of the term a^ h^ c^ dy As the elements
are arranged in their natural row order we have the series,
3, 2, 4, 1, from which to determine 'the sign. There are
three inversions and the sign is therefore — .

5. For the same determinant as in the last example find
the signs of the terms ^2 ^4 ^3 ^1 5 ^3 ^1 ^2 ^4 ' ^4 ^1 ^3 ^2'

6. Find all the terms of the determinant

^1

^2

^3

h

h.

^3

^1

^2

^3

290. Theorem. The iiumhei' of terms of a determinant of
the nth order is n !. This follows at once from the fact that
each term contains one constituent from each row and one
constituent from each column, and therefore there are
as many terms as there are arrangements of n things all
together.

291. Theorem. In any determinant there are as many posi-
tive as there are yiegative terms; for if we interchange two
suffixes of tlie second series in a positive term, we get a
negative term, and therefore there are as many or more
negative terms than there are positive terms, and the inter-
change of any two suffixes in a negative term will give a
positive term, and there are as many or more positive terms
than there are negative terms, and therefore there must be
the same number of each.

292. Theorem. If every element of a row or column of a
determinant is zero, the determinant is zero. This follows
from the fact that every term of the determinant contains
one element from this row or column.

DETERMINANTS 249

293. Theorem. Two determinants tvhich differ only in hav-
ing the rows of one the same as the cor responding columns of the
other are equal. Every term of the one determinant contains
an element from each row and each column of that deter-
minant, and therefore it contains an element from each
column and row of the other determinant, and therefore is
a term of the other determinant. That the sign factor of
these two terms is the same is evident, since it is determined
from the same series of numbers in each case.

From this it follows that in any proposition involving the
terms " row " or " column " we may get another which is
equally true by substituting the terms " column " or " row "
respectively.

1. Show that

EXAMPLES

«1

«2

H

^1

h

h

=

H

H

^3

a^ ^j c^

^2 2 ^2

a^ O3 c^

2. Write the negative terms of the determinant in the
preceding example.

294. Theorem. If tivo columns of a determinant he inter-
changed^ the resulting determinant differs only in sign from
the given determinant; for this amounts to an interchange of
two of the second set of suffixes in each term and therefore
to a change of sign of that term ; consequently the sign of
the whole determinant is changed. '

295. Theorem. Jf two roivs of a determinant he identical.,
the determinant is equal to zero.

Let the determinant be A, then interchanging these two
rows, we have a determinant which by the preceding article

250

COLLEGE ALGEBRA

is equal to — A, but the resulting determinant is exactly the
same as A on account of the identity of the two rows, there-
fore

A,

or

A =
A = 0.

^11

«12 •

•• «1«,

«21

^22 '

•• «2«

^nl

^«2 ••

^nn

296. Since by definition every term of a determinant
contains one and only one constituent from each row and
column, it follows that a determinant is a linear homogeneous
function of the constituents of any row or column. Thus

~ ^11^11 "I" ^12^12 + *^13^13 + '** + ^1«^1»'

where the ^'s contain no constituent from the first row.

297. Theorem. If all the constituents in any roiv he multi-
plied by the same number, the resulting determinant is equal to
the product of the original determinant and this number.

For if we expand the determinant as a linear function of
the elements in this row, the given number will appear as a
factor of every term and therefore of the determinant.

298. Theorem. If the constituents of any row differ from
those of any other row by the same constant factor^ the deter-
minant vanishes.

For taking out the common factor, there results a deter-
minant with two indentical rows, and this is equal to zero.

Example. In the determinant

2 5 6
3-3 9
14 3

DETERMINANTS

251

the elements of the hist column are three times the corre-
sponding elements of the first column and therefore

5

= 3

2 5 2
3-3 8
1 4 1

= 0.

299. Theorem. If each of the constituents of a row of a
determinant consists of tivo terms^ the determinant ynay he ex-
pressed as the sum of two determijiants.

For if the determinant be

A =

an + ^\i «i2 + ^^i2 ••• 'hn + K

«21

a,^i

22

^0

^n2

a,

expanding in terms of the elements of the first row we have,
A = («ii + ^i)^ii + (^^12 + ^12)^12+ ••• +(^i« + ^i«Mi«

= ^1^11 + ^12^2+ ••• + ^l«^l« + ^11^11 + ^12^2+ ••• +hnAn

^11

<^12 •

' (^m

^11

^^12 •

•• Ki

«2i

«22 •

• (l2n

+

«21

^22 ■

■' (hn

<^nl

<^^n2 '

' ^nn

f/„l

««2 •

■ ^nn

300. Theorem. If each of the constituents of a roiv is equal
to the sum of r terms, it is obvious that the determinant is
equal to the sum of r determinants. This may be farther
generalized by having polynomials for the constituents of
other rows.

301. Theorem. The result of 299 may be viewed as a
theorem for the addition of two determinants, giving the

252

COLLEGE ALGEBRA

theorem : If two determinants are alike except as regards the
elements in the rth row of each^ their sum is equal to a de-
terminant which is like each of the others except that any
constituent of the rth row is the sum of the corresponding
constituents of the two given determinants. This theorem also
may be generalized.

302. Theorem. If the constituents of any row he increased
(^algebraically^ hy equimultiples of the corresponding con-
stituents of any other row^ the determinant is unaltered.

For the resulting determinant is equal to the sum of two
determinants, one of which is the original determinant and
the other is this multiple times another determinant having
two rows identical, and therefore vanishes.

The principle of this article is useful for simplifying and
evaluating determinants whose constituents are numerical.
Thus if in the determinant

2 13

-3 -2 -4

we add the elements of the second row to those of the third,

we have

2 13

-3 -2 -4

0-1 1

and in this if we add two times the elements in the first row
to those of the second, we get

2 1
1 0
0 -1

DETERMINANTS

253

and finally if in this last form we add the elements of the
second to those of the third column, we get

2

1 4

1

0 2

0

-1 0

which when the common factor 2 is removed from the third
column is seen to vanish, having two identical columns.

303. EXAMPLES

1. Show that a -\- h -{- c is 'd factor of

a h h -\- c

c -\- a
a -\-b

9

a — mg d g

a d g

h

=

h — mh e h

=

h e h

k

c — mk f k

c f k

2. Show that

a + md d
h -i-rne e
c + mf f

3. Write in determinant form the following expressions

(1) ahc + 2 hgf - g% - ¥c -f^a,

(2) 3 xyz — Q^ — y^ — z^. ,

4. Show that

5. Prove

ho

a

ca

h

ah

e

a^

n1

2 1-9

8-5 4

12 7-3

a''

119

= 2

4 5 4

6 7 3

304. Minors. If we delete a rov/ and a column of a deter-
minant, the determinant of the remaining (ii — 1)^ elements

254

COLLEGE ALGEBRA

is called a minor of the (w — l)th order of the original deter-
minant, or a first minor.
From 296 it is seen that

A =

a,

11

21

12
''^22

'in

''2H

a

m

a

«2

a.

= ^.1^11 + ^10^10+ ••. +«i„4

"12^-^12

In^^lni

and that the expressions A-^^ A-^^^ •••, A-^^ contain no elements
from the first row. What these expressions are may be seen
on referring to the definition. For the coefficient of 6Kjj by
definition must contain one and only one constituent from
each of the other rows and columns except the first, and there-
fore can be nothing more than the minor formed by deleting
the first row and column. In the same way after passing
the second column over the first, which changes the sign of
the determinant, we find that A-^^^ the coefficient of a^^^ is the
negative of the minor obtained by deleting the first row and
second column ; similarly ^j 3, J.J5, ••• are the minors formed
•by deleting the first row and third column, first row and fifth
column, etc., and ^j^, ^jg, ••• are the negatives of the minors
obtained by deleting the first row and fourth column, the
first row and sixth column, etc.

If we denote by A^j, the minor formed from A by deleting
the ith. row and ^th column, and expand A in terms of the
elements of the ith row, we have

A=(-iy-V.Ai-«.2^2+ ••• +(-iy^-KA,j.

Using this principle to expand the determinant

A =

a

a.

a

12

ar

22
31 ^32

a,, a

a

"41

42

a

a

■13
23

^33
43

a

a

24
^34
44

DETERMINANTS

255

we have A = — «52i^2i + ^^22^22 "~ ^23^23 + ^24^24

= ^31^3^ — ^32^32 + ^'^33^33 — <^^34A34

= - ^14^14 + ^24^24 - ^^34^34 + ^^44^44'

Applying this to a numerical example, we have for the
determinant

1

2

3

2

5

4

3

7

8

5

2

6

1

6
5
3

5 4 6

2 4 6

2 5 6

2 5 4

=

7 8 5

-2

3 8 5+3375

—

3 7 8

2 6 3

5 6 3

5 2 3

5 2 6

5 4 6

5 4 2

2 4 6

2 5 61

7 8 5

—

7 8 3

-2

3 8 5+3375

2 6 3

2 6 5

5 6 3

5 2 3

5 4

4

2

7

=

7 8

2

+

3

5

2 6

-2

5

-6

—

-20.

(301, 302)

305. The expression

'^31^21 ' ^32 22 ■" '^33'^23 ■" ^34^24'

which is obtained from an expansion of the determinant A of
304, by putting the elements of the third line

in the place of

'31' '■*32' "'33'' ^^34'
^2V ^22' ^23' ^24'

respectively, obviously vanishes, since it is equivalent to the
determinant

'11

^31

^2 ^13

a

^32

'33

14
^34

^31 ^32 ^*^33 ^34

a

41

'i2

hs ^44

in which the second and third rows are identical.

256

COLLEGE ALGEBRA

Thus in general if we multiply the elements in any row of a
determinant hy the corresponding cof actors of the elements of
any other row^ the sum of the products thus formed is equal to

zero.

306. If we multiply the three equations
a^-^x + ay^y + ^132; = ^14,

a^^x + ^32^ + a^^z = ^34,

by A^-^, ^21' ^3P resjDectively, and add them, we have
(^11^11 + ^21^21 + ^31^31)^^ + (^12^11 + ^22^21 + ^32^31)^

+ («13^11 + «23^21 + ^33^31)^ = «14^11 + ^24^21 + «34^31-

If we denote by A the determinant

11 1^

13

a.

21

a

a

31

23

^32 ^33

whose elements are the coefficients of x, y, z, in the three
given equations, it will be seen that the coefhcient of x in
the fourth equation is A, that the coefficients of y and z
vanish, and the right-hand member of the equation is the
determinant formed by writing a^^, a^^, a^^, in place of the
elements «j^, a^i^ a^^ in A.
We have therefore

x =

«14

^12

«13

^24

^22

^23

%4

'^32

^33

DETERMINANTS

257

Similarly

y =

z =

%1

'14

24

a

34

^13

•23

a

33

A

^^11

«12

^^14

«21

«22

^^24

«31

^32

%4

SO that in general three linear non-homogeneous equations
in three unknowns have one and but one set of values of
X, y^ z which satisfies them.

EXAMPLES

1. Solve the equations

X — \y = ^.

We have

x= -

5

3

(3

-4

2

3

1

-4

— 3 8.
""" 11'

and

y =

2

5

1

G

2

3

1

-4

7_

~" 11'

2. Solve the equations

2a;-3^ + ^ = 7,
x-\y-\-±z = ^,
3a;-f-y— 3s = — 4.

258 COLLEGE ALGEBRA

307. If in the equations of the preceding article a-^^ = a^^ =
a^^ — 0, and therefore the equations are homogeneous, it fol-
lows that if we are to have values of x^ ?/, and z other than
zero, which satisfy the equations, A must be equal to zero.
That is, to have a set of three linear homogeneous equatio7is in
three unknown quantities^ satisfied hy values other than zero^ the
determinant of the system must vanish.

This is evidently just as true for n linear homogeneous
equations in n unknowns as for three.

Example. Show that the three equations
ax+{h + c)y-\-z^^,
hx-\-(c-\- a^y + 3 = 0,
ex + (a + J)?/ + 3 = 0,
are satisfied by values of x^ ?/, z other than zero.

308. If we divide the three linear homogeneous equations
of the preceding article hj z (z4^ 0), and put

X

- = u,

z

and

z

we get

a^^u + a^^v + fl'i3 = 0,

a^^u + a^^v 4- a^^ = 0,

a^^u + a^^^v + «33 = 0,

that is, three equations with two unknown quantities, u and v.
But two independent non-homogeneous linear equations de-
termine a unique set of values of u and v^ and there is no
third equation which is satisfied by these same values, unless

DETERMINANTS

259

it is dependent upon the other two equations. Solving the
first two for u and v and substituting these values in the
third, we get A = 0, wliich coincides with what we have just
seen, viz. tliat A must vanish in order to have the three equa-
tions consistent. In fact A = 0 is satisfied if we take

ttq9 — ^19 ~T~ '^^^22'

a = a-,o-\- \a

^^33

13

^23'

that is, the third equation is

^11

u + a^^v + rt^3 H- \(^a.2{ii + ^22^ + ^23) == ^'

or it is equivalent to the first plus X times the second ex-
pression equated to zero.

309. Product of Two Determinants. Let

A" =

a

11

'21

12

22

'13

•^23

^31 ^^32 ^33

, A' =

'11

^12 ^

^21 ^^22 ^
h

13

23

'31

'32

33

^11^11 + ^12^2 + ^13^3 ^21^^11 + ^^22^12 + ^^23^13
^11 21 ~^ ^^12^22 ~^ ^13 23 ^21 21 ~^ ^22 22 "•" ^*23 23
^11^31 + ^12^32 + ^13^33 ^'^21^31 + ^22^32 + ^23^33

^3Al + ^^32^12 + ^^33^3
^31 21 •" ^32 22 "■ ^33^23
%1^31 ■^" ^32^32 + ^33^33

The determinant A^' may be partitioned into twenty-seven
determinants of the third order with monomial elements.
Twenty-one of these determinants vanish identically, having,

260

COLLEGE ALGEBRA

after common factors are removed, two or more identical
columns. Thus one of them is the determinant

^11^11 %l*^ll ^^32*^12

11 91 ^91 91 ('oc}0()(f

^11^31 ^21^31

= a

11

a.

21

'32

^11 ^11
^21 ^21 ^^

'12

22

'31

'31

32

= 0.

The determinants which do not vanish identically are the
following :

^11^1

^22 12

%3'^13

^11^11

^23-13

^32^12

'^11^21

^33*^23

1

^11^21

^23^23

^^32 22

«11^31

^22 32

<^33''33

^11^^31

^23^33

''*^32 32

%2 12

^21^11

^33*^13

'^12^12

^23^13

hAi

^12 22

^21^21

^33^23

1

^12 22

^23 23

HxK

^^12 32

^21^31

^33^33

'^12*^32

^23''33

'hAi

«13^^13

%1^11

^*32'^r2

^13-13

^'I'Pvi

^31^11

'^13^23

^21^21

^^32 22

^

^^13^23

^22 22

a^iO^i

^13^33

^21^31

^32^32

^13^^33

^22 32

^31^31

Talking out the common factors, we have

A = A '^ii^22'^33 ^ ^11*^23^32 ~ ^ ^12*^21^33 "• ^ '^12^23^^31

+ A'a^^a.^^(u^ - A^)'igrt22% = AA'.

It will be observed that to form a determinant which does
not vanish identically, the first column may be chosen in
three ways ; and when tliat is chosen, the second may be
chosen in two ways, and the third in one way, making in all
six determinants which do not vanish identically.

The student may state the rule for the multiplication of
two determinants.

DETERMINANTS

261

Example. Write as a determinant the square of

X

•/-r

1 -^2 -^3

yi Vi Vz

^1 ^2 ^3

310. EXAMPLES

1. Evaluate the determinant

3

4

6

7

5

4

9

8

1

2

7

3

0

5

3

0

2. Expand

a h

0

h h

f

g f

c

3.

Expand

0

a h e

a

ode

h

d 0 f

c

e f 0

Expand in terms of the
elements of the 4th row.

4. Evaluate

4

9

2

3

5

7

8

1

6

The numbers in this determinant are arranged in what
is known as a magic square. The sum of the elements in
any line is 15.

5. Solve the equations

x + 2y + z = A,

5x— 1/ — oz = l^

4a: + ^ — 2 = 3.

CHAPTER XVIII
THEORY OF EQUATIONS

311. We have now to consider the general properties of
the polynomial with real coefficients,

f(x) = a^x"" + ^ia^"~i + ^2^""2 + ••• 4- a^''-^ + ••• + «,„ (1)

and those values of x which make it vanish.

312. A question which naturally arises concerning the
polynomial is, which are the significant terms in the case
of large or small values of x.

In order that

a^x^ > a^x^~'^ + a^x^~^ + • • • + a^, we must have

a-,x^ ^ -{-■'• -\- a.

Dividing numerator and denominator of the left member of
the inequality by x''~'^, we must have

^0^1 __._ -> i

Li, f> t*^.

a. +^+ ••• +
^ x x

,11— \

(3)

313. As X becomes indefinitely great, the left member of
the inequality becomes indefinitely great, and therefore for
some value of a;, as x increases, the fraction must be greater
than one ; that is, as x increases there is a value of x for
which and for greater values,

a^^x^ > a^x^~'^ + a^x^~'^ + [- a^, (4)

262

THEORY OF EQUATIONS 263

314. By what has just been proven,

for sufficiently great values of y, that is

a^ «^ «5 (6)

y y y

or II - = a;,

y

for sufficiently small values of x. Therefore the independent
term is greater than all the others if x is taken sufficiently small.

315. The series 2_i_ _i_ n

may he made as small as we please hy making x sufficiently
small, for it may be made smaller than any assigned number,
however small, which may be taken as a„.

316. In the same series the sign of the series may he made

to depend upon that of the term a^-^x^ for the series may be

written . „_j.

X\a^_-^ -f- aj^_^ -f- ••• + a^ j^

whence the statement is evident.

317. Development of a Function. If in /(a;) we put x-{-h
for x^ we have

f(x + A) = a^^x + hy + a^Qx + A)"-i + ••• + a„_i(x + A) + a„
= a^x" + a^x''-'^ H \-a„+\ ^la^x''-'^ + {n — l^a^x""'^

+ ••• + 2 a,,_^x + a^_-^lh + ^ ln(n - l^a^x^-^- + 0^ - l)^^ - 2)

«i:r"-3 4- ... + 3 . 2 . a,,_^x + 2 a,Jh'^+ _^ n^i - 1) ..-l ,^.

n !

264 COLLEGE ALGEBRA

or denoting the coefficient of Jf by - — ^-^,

fCx + h) =fix) + hfix) + ^J"ix-) + ... + ^n-Kx-) .

318. The different expressions/'(a:),/''(2;), etc., are called
the^rs^, second^ etc., derived functions of x. It will be seen
that /'(a;) is formed from /(a;) by multiplying each term in
f(x) by the exponent of x in that term, diminishing the
exponent (of x^ by one, and taking the sum of all such
terms. Again /"(a;) is the first derived function oif'(x')^
and in general /^^^ (a;) is the first derived function oif^'~'^\x^.

EXAMPLES

1. Given f(x) =^ x^ -1 x^ + x-1, find fQx + 2).
We have for the derived functions

f'Qx) = S)x^-4:x + l,
f"{x-) = lSx~4,
f"'(x) = lS.

Using X for 7i, and 2 for x in the foregoing formula, we have

/(2) = 17,

/'(2) = 29,

/"(2) = 32,

/"'(2)=18.

Therefore f(x+2} = 11 + 29 x + 16 x^ + S x?.

2. Given /(a:) the same as in the last example, find

f(x + l),f<:x-2-),fCx-l).

THEORY OF EQUATIONS

265

319. From the above development for/(a:+ li) we get
fix + 70 -/(x) = hf'ix) + ^/" (:,) + ... + ^ /W(^).

By 316, /(:2:^+ A) —f(x) may be made less than any assigned
number by taking h sufficiently small. This means that if x

Fig. 27.

increases by indefinitely small increments, from x^ to x^^ /(^)
changes by indefinitely small increments from f(x^ to f(x^)^
or f(x) is said to vary continuously
between x^ and x^. As a conse-
quence of this, it is readily seen
that if f(x^ and f{x^ have oppo-
site signs, there must be some value
of X lying between x^ and x^^ such
that/ (a:) vanishes for this value.
The graphical significance of con-
tinuity is that the curve y—f{x^
is uninterrupted between
any two of its points as in
figure 27, and cannot be
interrupted as in figure 28. Fig, 28.

266 COLLEGE ALGEBRA

320. Theorem. Every function of odd degree vanishes for
a real value of x which has a sign opposite to that of a^, for by
substituting — oo, 0, and +qo for a;, the function is negative,
has the sign of a„, and is positive respectively, and therefore,
by 319, vanishes for a value of x opposite in sign to that of
a„, for the function will change sign between — go and 0, or
between 0 and +go according as a^ is positive or negative.

321. Theorem. Every function of even degree which has a^
negative vanishes for one positive and one negative value, for
this function changes sign between — oo and 0, and between
0 and + Qo . This is seen graphically in that to change sign
the graph representing the function must cross the axis of x,
hence there must be a root for the value of x at which it
crosses.

EXAMPLES

1. Show that x^ — 2x'^-{-5=0 has a negative root.

2. Show that x^— 2x^ + 4: x^— Sx — 2 = 0 has at least a
positive root and a negative root.

322. In 81 we have shown that if a^^ 0,

f(x) = a^^x - ccj) (x-a^} -'• (x- a„),

where a^, «2i ••• «„ are the roots of /(a;) = 0. It may hap-
pen that these factors are not all different ; for instance,
(a: — otj) may occur once, twice, or any number of times up
to n times, in which case «j is said to be a double, triple, etc.
root of the equation. The equation is still said to have n
roots. How a double root forms the transition between two
real and distinct and two conjugate imaginary roots has
been shown in 77.

THEORY OF EQUATIONS 267

323. Descartes' Rule of Signs. Concerning the equation
obtained by placing f(x) = 0, Descartes gave the following
rule in regard to the roots : The number of positive roots of
an equation cannot he greater than the number c of changes of
sign in passing from the first to the last term^ and the number
of 7iegative roots cannot be greater than the number c' of changes
of sign inf( — x')= 0,

When/(ir) = 0 is a complete equation, that is, an equation
which contains terms involving all powers of x from aP to x"'
inclusive, if p denote the number of permanences of sign,
c -{- 2^) = n, p = e\ and therefore c -{- c' = n, but if the equation
is not complete, c' is not necessarily equal to p.

When imaginary roots exist, it is often possible to de-
tect their presence by Descartes' rule of signs. For when
c -{- c' <n^ that is, when the number of positive and negative
roots together is less than n, the whole number of roots of the
equation, then n— (^c -\- <?') is an inferior limit of the number
of imaginary roots.

EXAMPLES

1. Show that the equation x^— 2x^-\-4x^-Sx—2 = 0
cannot have more than three positive roots and one negative

root. The series of signs is H 1 — , in which there are

three changes from + to — or from — to +, hence there cannot
be more than three positive roots. If x is changed into — x^

the signs become -\ — f- + H , where there is but one change

and hence there cannot be more than one negative root.

2. Show that the equation a;* — 3 a; + 1 = 0 has at least
two imaginary roots.

3. Show that the equation 2 a;* + 5 a:^ + 3 = 0 has all its
roots imaginary.

4. Find the upper limit of the number of positive and
of negative roots in the equation x^ — Sx^-\-x— 1 = 0.

268 COLLEGE ALGEBRA

324. Complex Roots enter Equations in Pairs. If a-{- pi
is a root of /(a;), a — /3z is also a root. Substituting
a -|_ pi for X in f(^x) and collecting the real and imaginary
parts separately, it takes the form /(« + /3i) = A -\- Bi^
which by hypothesis vanishes, that is A + Bi = 0, therefore
A = B = 0. Since a — (Bi may be obtained from a 4- Pi by
changing the sign of ^, it is seen that fQi — pi) = A — Bi=0,
since A and B are zero ; therefore a — pi is a root of

By a precisely similar process it can be shown that if
^-^ V^ is a root, a — V/^ is also a root of /(a;) = 0; that is,
binomial quadratic surds enter equations in pairs as conjugates.

Example. The equation rr* — 4a;^ + 4rr — 1=0 has 2 + V3
for one root. Find the other roots.

325. Theorems. If a and h are tivo numbers of which b is
the greater, and f(a) and f(b) have the same or opposite signs,
then an even or an odd number of real roots of f(x) = 0 lies
between a and b.

For convenience of statement zero has been included
among the even numbers, for it is apparent that there may
be no real root when /(a) and /(J) have the same sign.

Let ce^, «2' "31 ••> ^r ^^6 ^11 ^1^6 ^Gal roots of f{x) which lie
between a and b. Then

f(x) =(x- a^)(x - «2) ••• (^ - f^,>)F(x),

where F(^a) and F(b') have the same sign ; for if they had
different signs, there would be a root of F(^x) and therefore
another root of fQc) besides a^, a^, •••, a^ lying between a
and 6, which is contrary to hypothesis.

THEORY OF EQUATIONS

269

Substituting a and h for x^ we have

/(a) = (a- a^{a - a^) .-• (a- Ur^F^a).

0)

/(^) = (^-«i)(^-«2) - Cb-ar)Fiib).

In /(«) the factors (a — «j),
• ••, (a — a,,) are all negative,
while in f(b) the factors
(6 — ccj), •••, (6 — «,,) are all
positive ; if /(a) and /(^)
have the same or opposite
signs, the products (a— ctj).--
(« — «,.) and (^ — ctj) •••
(5 — a^) have the same or
opposite signs ; but the
product (5 — «^) ••• (^ — «,.)
is always positive, therefore
the product (a — «^) •••
(a — «^) is positive or negative
odd. Therefore when /(a) and

(2)

Fig. 30.

Fig. 29.

according as r is even or
/(^) have the same sign,
the number of real roots r
lying between a and b is
even (or zero), and when
/(a) and f (by have opposite
signs, the number of real
roots lying between a and
h is odd.

Conversely, if an even
nu7nber of real roots of f\x)
lies between a and ^, /(a)
and f(b) have the same
sie/ns, and if an odd number

270 COLLEGE ALGEBRA

of real roots lies hetween a and 5, /(«) and f(h) have op-
posite signs, as can be seen from the expressions (1) and
(2) for f(a} and f{b). Graphically, we see, if there is a
change of sign, the curve representing f(x^ must cross
the axis of x an odd number of times, as in figure 29 ; if
there is no change, an even number of times, as in figure 30.

EXAMPLES

1. Prove that x^-\-S a^—60 x^-\-2 x-^l = 0 has a positive
root between 0 and 1, and a negative root between 0 and — 1.

2. Show that the equation x^ — S x^ -\- x —1 = 0 has at least
one positive root between 1 and 2.

326. Since
f(x) = a^x"" + «^a:''~i + h «„ = a^(^x — a^ (x— a^'-'(x— «„),

•^ ^x^^-'hx''-^ + ... + ^ = (:i; _ «^)(^ _ ,,^) ... (^ _ ^^),

we have, using p^ for ~^,

%

P\ =-(«i + «2 + ••• + 0=-2«i,

P2 = (V2 + ^^i«3 H 1- ««-!««) = ^«i

2'

Fs = ~ ^"i

«oO£

2^31

Pr= (-l)'"2«i«2

i^«=(-l)''-«l«2

a.

For, to get the coefficient of a:"~^ in the product, x must be
selected from (^n — r) factors and as from the remaining r
factors in all possible ways.

THEORY OF EQUATIONS 271

327. Cube Roots of Unity. To find the roots of x^ = 1, we
have Q(^ — \ = (x— V)Qx^ + a; + 1) = 0 ; the roots are therefore
x=l and a:= — |±|V — 8. If we denote the imaginary
roots by &> and w', it is easily seen that (o' = (o^, o) = w'^, and
(o(o' = 1, from which we see that the three cube roots of unity
can be expressed as 1, o), w^ and that 1 -{- co -{- (o^=0.

EXAMPLES

1. x^-\-7/^=(^x-\-^^(^0)X + co'^^)((o'^x-\-(O7/^. 0)= — -|-4- JV — 8.

2. x^ — y"^ — {x — y~){wx — co^y^iccP'x — (oy).

3. {x-{- (Dy -\- ccr'z)(x-{- ccP'y-\- coz) =x^-\-y'^-\- z^— xy — yz — zx.

4. (x-\-y-\- z')(^x + Qx^y + coz^ (^x + coy-\- «%) = a^ -\- y^ -{- z^

, . — 3 xyz.

5. (l-ft))3=_3ft)(l-w) = -8V^=^.

6. (l-a))2 = -3a).

7. li x-\- y = u^, (OX + M^y = ti^, ap'x + &)?/ = 7/3, then
{u^ - u^)(u^^ - '^3)0^3 - u^) = - 3V- 3 (x^ - ?/3).

SYMMETRIC FUNCTIONS

328. Definitions. The relations between the coefficients
and the roots given in 326 enable us to express certain
functions of the roots in terms of the coefficients without
knowing the values of the individual roots.

A function is symmetric with respect to its variables when
the interchange of any two of them leaves its form unaltered.
Thus X + y -{-z^ 2 x^ -{- 2 y^ -\- 2 z"^ -{- S xy -\- S yz -{- S zx are sym-
metric and are denoted by Sa:, 2 ^x^ + 3 ^xy.

The functions of the roots contained in the relations of
326 are known as the fundamental symmetric functions. By
means of these relations every symmetric function can be
expressed in terms of the coefficients.

272 COLLEGE ALGEBRA

If a, /3, 7 be the roots of x^ -\-px^ -\- qx -\- r = 0^ we may find
the symmetric function Sa^ as follows :

^''''''' (« + /3 + ./)2=«2+^ + ^2 + 2«/3 + 2;e7 + 2 7«,

we have '^a^= (Xa)'^ — 2 Xaff = p^ — 2 q.

To find Xa^/3 we have

(«yS + /37 + 7a)(« + yg + 7) = 2^2^ + 3 afiy.

Therefore Xa^^ = —pq 4- 3 r.

. ^2 + ^^2+2 2_^«2

ap P7 7«

^ 7(«2 + ^2)^,,^^2_^^2)_^^(^2^^2>)

«/37
— ;?gH- 8 r

— r

EXAMPLES
1. For the cwhio, x^ -{- px^ -\-qx-\- r=(i^ find the symmetric
functions ^a^ ; ^a^/3^; (« + ^)(^ + 7)(ry + «) ; 2 "^ "^ ^^ ;

a

j3 y a a /3 y

2. For the biquadratic a;^ 4- Jt>a;3 + ^.^2 -^ ^2: + s = 0, find
2«2^2. 5;,,2^^.

329. Factoring of Symmetric and Related Expressions.
From the definition of a symmetric function it is apparent
that the sum, difference, product, and quotient of two symmetric
functions are themselves symmetric.

THEORY OF EQUATIONS 273

The function (^a—hx)(h — cx)(^c — ax) is unchanged if a
is changed into 6, h into <?, and c into a, whereas, if we inter-
change a and 5, its form is changed. Though such a func-
tion is not symmetric, it is said to be cy do -symmetric.

The function aQ)^ — (P') + h(^c^ — a^) + ^(a^ _ 52^ jg changed
into its negative when any two of the letters a, 5, c are inter-
changed. Such a function is said to be alternating.

The sum., difference., product., and quotient of two cyclo-sym-
metric functions are cy do -symmetric functions.

The sum a7id difference of two alternating functions are al-
ternating functions, and the prodiict and quotient of two alter-
nating functions are symmetric functions, the same variables
being involved in each case.

The preceding propositions, together with the factor theo-
rem, make it very easy to factor certain symmetric and
alternating functions.

EXAMPLES

1. Factor a(P - (P)+ b(^c^- a'^)-{-c(ia^ -b'^^.

Using the factor theorem, first trying for monomial factors,
by putting <x = 0, we see that this does not make the expres-
sion vanish and hence a is not a factor. Cyclo-symmetry
then shows us that neither b nor c are factors. Next try
for binomial factors by putting a = b. In this case the
expression vanishes and therefore a — 6 is a factor, and cyclo-
symmetry shows us that b — c and c — a are also factors.
The expression being of the third degree, there can be no
other literal factor, hence

Since this is an identity, we must liave

ab''-=Nab'^,
and therefore iV= 1.

274 COLLEGE ALGEBRA

2. Factor a^ (h — c) -\- h^ (^c - a) + e^ {a - h) .

This expression is a homogeneous, alternating function of the
fifth degree. As before, we find that a—h^ b — c, and c — a
are factors, and therefore (^a — h^(h — c)(^c— a)^ which is an
alternating function, is a factor, therefore the other factor,
which is the quotient of the given expression by this prod-
uct, must be a homogeneous symmetric function of the
second degree. Hence,

a\h - (?) + h\c - a) + c* (a - 5) = (a - h^(h - e)(^c - a)[iV(a2

+ h'^ + 6'2) + M(iah + hc^-ca)'].

Since this is an identity, we have, equating coefficients,

a^h = - Na%

and therefore iV= — 1.

Similarly, a%W- a^'^M^ 0,

or iV^= M,

The coefficients il!f and iV might have been found by giving
special values to a, ^, c. Sometimes it is more convenient
to use one method and sometimes the other, or in many cases
it is most convenient to combine the two. Thus, after hav-
ing found N by equating coefficients, we might find M by
substituting in the identity 0, 1, — 1 for a, 5, c^ respectively,
and obtain

(- 1) + (-!) = (- 1)(2)(- 1)[- (1 + 1) + Mi- 1)],
from which we get iltf = — 1.

Hence,
a^(h-c) +^*(^-^) -\-o^(a-h)

= — {a — b){h - c)(^c - a'){a'^ -\- b"^ -h c^ + ab + be + ca).

THEORY OF EQUATIONS 275

330. To form the equation whose roots are the roots of the
equatio7i f(x) = 0 each multiplied hy a constant factor A, we
have, if a be any root of the equation f(x) = 0, yg = ha as the

B
corresponding root of the sought equation, whence a = - , and

therefore, since /(«) = 0, we have/[y 1 = 0, and so for every
root, and hence if y represents the unknown of the sought
equation, it is fi'^\ = 0. lif(x) is the polynomial

a^x" + «i2:""i _|_ ... 4- ^^^
we have /?/V , fv\~'^ , , a

or multiplying by h'\ the equation in y becomes

a^yn + A^i^^-i + h\y''-^ + ■" + h"a,, = 0,

in which it is seen that the coefficients are formed by multi-
plying aj. by h'' for all values of r from 0 to n.

By this means any equation with fractional coefficients can
be transformed into one in which a^ is unity and all other
coefficients are integral. Thus if we multiply the roots of
theequation ^4 + j^ + |^a + |^+ 3 = 0

by 30, which is the least common multiple of the denomina-
tors of the coefficients, we have

x^-^lb3^-{- 300 x^ + 10800 X + 2430000 = 0.

Sometimes a smaller number than the least common multi-
ple of the denominators will serve to free the equation from
fractional coefficients.

If in the transformation of 330 we make A = — 1, the
resulting equation becomes one whose roots are the negatives
of those of the given equation.

276 COLLEGE ALGEBRA

331. To find an equation whose roots are the reciprocals of
the roots of the given equation^ we have

/(^) =/(-) = 0,

or ?a + -^+ ... +a„=0,

whence by multiplying by y^ we have

and we see that the coefficients of the resulting equation are
formed from those of the given equation by changing a^ into
a„_^ for all values of r from 0 to n.

Example. Find the equation whose roots are the recip-
rocals of the roots of the equation

332. To form an equation whose roots are the roots of the
given equation diminished by A, we have, if ?/ be a root of the
proposed equation, y — x — h^ and therefore x = y -\-h^ hence
f(x) —fiy + A) = 0 is the required equation in y. The iden-
tity / (a;) =/ (y + 70 gives

or \i y be replaced by its equal x — 7i, we have

a^x^+a^x''-'-^- ... +a,, = AQ{x-h}n-^A^(x-hy-'^+ ...

■i-A^_^{x-h)+Ar,.

If we divide both sides of this identity by x — h, the remain-
der will be the value of the coefficient A„. If we divide the

THEORY OF EQUATIONS 277

resulting quotient by x — h, tlie next remainder will be A,^_^,
and by continuing this process all the ^'s may be found. It
will be observed from this that to obtain the coefficients
in the required equation we have but to make repeated
applications of the remainder theorem.

333. If Q is the quotient and R the remainder (indepen-
dent of x) when /(a;) is divided hj x—h^ we have

where .§= ^Qa^^-^-f- V^'^ + ••• + ^.-i-

Equating coefficients of like powers of x on both sides of the
equation we have the following relations,

or, as they may be written,

bQ = aQ, b^ = hbQ + a^, b^ = hb^-\- a^, •••, 5^= hb^.^-\-a^, •••,

R = hb^_^-{-a„,

from which we see that the work may be arranged as follows :

0 1 2 * ' * ^

hb^ Jib^ ••• hbn-i

h ^1 h ^

The coefficients of /(a;) are written in a row, zero coefficients
being supplied where necessary. The coefficient a^ is brought
down as b^. This is multiplied by h and added to a^ giving
by This in turn is multijDlied by h and added to a^, giving

^2, and so on.

It Avill be observed that this process gives both the coeffi-
cients in the quotient and the remainder when /(a:) is divided
by a:— h. If the remainder is zero, /(a;) is exactly divisible
by a: — A, or has x — h as ^ factor.

278 COLLEGE ALGEBRA

EXAMPLES
1. To diminish the roots of the equation

by 2, we have

1

-3

2

5

2

-2

0

1

-1

0

5

2

2

1

1

2

2

Hence the required equation is

x^-\- Sx^ -{- 2x -{- 5= 0.

To increase the roots of an equation by h would be the same
as to diminish them by — h.

2. To increase the roots of the equation
x^-\-Sx^-\- 2 2^ + 5=0,

by 2, we have

1

3

2

5

-2

-2

1

1

0

5

-2

2

1

-1

-2

2

1 -3

and hence the required equation is

a^-Sx^ + 2x-^5 = 0,
which is, as it shouki be, the equation of Example 1.

THEORY OF EQUATIONS 279

334. EXAMPLES

Find the successive derived functions of :
1. x^-2x^-{-Sx-l. 2. 2x'^-:i:^-h2x'^-\-5x+l.

3. x^ — ^ x^ -\- 7?' — 5.

4. \if(x) = 2:3 + 3 :r2 - 7 :r + 2, find/(2: + Ji).

5. \if(x) =x^-2 x^ + a; + 1, find/(2: - 2).

6. Show that the equation x^ -\- Z x^ -\-l x = ^ has no real
root except zero.

7. The equation x^ — :j^ — ^ x^ -\-W x -\- b = ^ has a root
a; = 2 + V— 1. Find the remaining roots.

8. One root of the equation a^* — 6 a:;^ + 13 a;^ — 8 2: — 6 = 0
is 1 + V2. Find the remaining roots.

Find for the biquadratic a;* + 'p:i^ + qx^ + ra; + s = 0 :

9. 2«2/37. 10. 2«3/3. 11. 2a*.

Factor :

12. (^ - 2)5 + (2 - 2^)5 + (2; - ?/)5.

13. a5(^ — c) +^^(^ — «) + <?^(rt — ^).

14. x\y^ - ^2) + if(z^ - x^) + 2;*(a;2 - j/2).

15. (a; + ^ H- 2;)^ — a:^ — ^3 — 2^.

16. {a-\-h-\- cy —{h-\- cY- (c + ay - (a + 5)* + a* + ^^ + c^-

17. Transform the equation x^-\-^a^ — ^x^-\-^x + l = 0
into an equation in which the coefticient of a;* is unity, and
all the remaining coefficients are integers.

18. Transform 3 a:* — | a:^ ^ 7 ^2 _ 1 ^ ^ 2 = 0, (1) into an
equation with integral coefficients, (2) into an equation in
which a^ = 1, and the remaining a's are integers.

19. Find the equation whose roots are the reciprocals of
the roots of the equation 2a:5— 3 a:* — 4a;^H-a:2_52;_j_7-_o.

280 COLLEGE ALGEBRA

Factor and solve the following equations by the method
of 333 :

20. a;3-9a:2+26:r-24 = 0, thus

1 _ 9 + 26 - 24 [2

2-14 24
1-7 12 0

Hence x—2 is a factor, the quotient or other factor is
x^—lx-\-12; proceeding with this as before we have

1_74-12L3

3-12
1-4 0

We see that a; — 3 is a second factor, and the quotient or
remaining factor is a: — 4. Hence, as in 84, we have

x^-9x^ + 2Qx-24: = (x-2)(ix-S)(x-4:),

Hence the roots of

^3_9^2_f_ 26a;- 24 = 0, are a; = 2, a;= 3, :r= 4.

21. x^-Sa^-10x-^24: = 0.

22. x^ ~ x'^ — 4:x — 6 = 0.

23. a;*-29a;2 + 100 = 0.

24. x^-4a^-Sia^-\-16x-{-105 = 0,

25. Diminish the roots of the equation

a:3 _ 2 a:2 + 5 a; - 3 = 0, by 3.

26. Increase the roots of the equation

2 a:3 + 4 a:2 - 3 a: + 5 = 0, by 2 .

335. The Geometric Interpretation of f'^Jr'). Let P andP'
be two points on tlie curve j/ =f(x)^ whose abscissas are x
and a: + A, respectively.

THEORY OF EQUATIONS
Y

281

Fig. 31

Then we have

MP'

h

that is,

h

= tan (/), (317)

or /W + A /'^(^) + A////(^) + ... = tan 0.

2!

3!'

Now as h = 0, P' = P, and PP' approaches the tangent
PT to the curve at P, (^ = r, and we have

f'(x) = tan T. (315 and theory of limits)

That is, the first derived function represents tlie slope or the
tangent of the angle ichlch the tangent Ihie makes with p)ositLve
direction of the axis of x. When /'(.r) is positive, /(a:) is
increasing, and conversely. When /'(a;) is negative, /(a;) is
decreasing, and conversely.

282

COLLEGE ALGEBRA

EXAMPLES

1. Find the slope of the tangent to the curve y = x^ — ^ x^
— 1, at the point (2, — 5). Here f(x^ is x^ — Zx^ — 1 and
f'(x) = 3 a;2 — 6 2^, which for the point (2, — 5) is zero.

2. Find the slope of tangent to the same curve as in the
last example at the points (3, — 1), (0, — 1), (1, — 3).

336. Rolle's Theorem. Between two values^ a and h (h >a),
of X for which f\x^ vanishes^ there is at least one value of x for
which f\x) vanishes. For since /(a;) is continuous between
a and 5, it must first increase and then decrease, or first
decrease and then increase, at least once, as x passes from a
to h ; that is, /' (x) must change from positive to negative or
from negative to positive and therefore must pass through
zero, once at least, as x passes from a to b.

It is to be observed that we have assumed in this proof
that /(:c) and /'(a;) are continuous and single-valued. The
accompanying figure illustrates the proposition.

>X

Fig. 32.

THEORY OF EQUATIONS 283

337. From the preceding theorem it follows that between
two consecutive roots of f (x} = 0 there may he no real root of
f{x) = 0 and there cannot be more than one such root.

338. Theorem. If (x — ay is a factor of f(x)^ then
(x—ay~^ is afactoroff'{x).

We have
/W ^fia+ix-a)^ ^fCa} + (:r- «)/(«) + -

^ Cx-ay-\p-i\a) ^ (x-ayf^^^(a') ^

(r — 1) I r ! *

But by hypothesis f(x) = (x — ayF(x^), and since these
values of f(x) are identically equal, the coefficients of the
corresponding powers of (x — a) must be equal, and there-
fore/(«)=/(«)= ••• =/'-i^(a) = 0. That is, a;- « is a fac-
tor not only oif(x) but of its first (r — 1) derived functions.

Similarly we have

(r - 2) !

(x- ay-lf^'\a) _(a^-ay-if(>Xa)

(r-1)! O'-l)! '

from which we see that (^x — ay-^ is a factor of /'(a;), and in
the same way we see that for the other derived functions we
have a corresponding result giving us the theorem. If a
occurs r times as a root of f(^x')=0, it ivill occur r—1 times
as a root off'(x)= 0, r — 2 times as a root of f" (x') = 0, and
so on to once as a root off''~'^\x^ = 0.

The development oif(x') shows conversely that if

/(a;) contains (x — a) r times as a factor ; that is^ a occurs r
times as a root of f{x^ = 0.

284 COLLEGE ALGEBRA

If/(x) = Ohas a root repeated r times, that is, a multiple
root, it can be found by determining the highest common
factor oi f{x) and /'(a;).

Example. Find the multiple root of the equation
Here /(^) =x'^ — x^—%7?'-{-bx—^^

f^\x) = 1\x-^,

from which we see that for a; = 1, /(:r), f'(x)^ and /''(a;)
vanish, and /(a:) contains (x—V) three times, /^ (a;) twice,
and /''(a;) once as a factor. In fact, we have

/(^)=(:,_ 1)3(^+2),

/(a:) = (.T-l)2(4a; + 5),

f"{x) = (x-l)(12x + Q).

339. Theorem. As x passes through a root a of f(x) = 0,
f(x) and f {x) have unlike signs for a value of x sufficiently
near and less than a and like signs for a value of x sufficiently
near and greater than a.

We have /(« + li) = hf („) + 1!/" („)+.. .,

/'(«+ A) =/'(«) + ¥"(«)+ -,

and since the first term of the right member determines the
sign in each development when h is sufficiently small, we see
that /(« + A) and /'(a + A) have opposite signs when h is
negative and like signs when h is positive.

This theorem is still true if « is a multiple root oif(x) — 0.
When a is not a multiple root of the equation, the truth of
the theorem is easily seen from the accompanying figure.

THEORY OF EQUATIONS

285

>x

Fig. 33.

Example. Using the equation of the example of the
preceding article, we have

fix) = {x- Vf(x + 2),
/'(a^) = (:K-l)2(4x + .5),

and we see that for a value of a: a little less than 1, f{x) is
negative and /'(a:) is positive, while for a value of a: a little
greater than 1, both are positive.

340. To transform a given equation into another having one
term less. Let us write for convenience /(a;) in the form

/(a:) = a^x^' + na^x'^--^ + '-^-%y^«2^"'^ +•••+«„.

Then

/(y + h) = r„y" + nv,r-' + ^^^1=^ r^"-^ + - + K

286 COLLEGE ALGEBRA

where FQ = aQ,

Fj = a^h + a^,

F2 = «Q/i2 + 2 a^li + «2'

F; = ajf + mi7i'-i + ^^"^""^^ ajf^ + • • • + ^..

^ 1

If in this expression for /(?/ + Ti) we wish to have any
one of the terms, say (^)FJ-^""% vanish, it is only necessary to

use for h one of the values for which V,. vanishes. Thus if
we wish to transform the cubic a^x^ + 3 a-^x^ + 3 a<yX + a^ = 0,
into one lacking the second term, we have J\ = aji + a^, or

h= i, so that the cubic in y becomes

<

«o2/' + -^-^-s — —y^^^^ ^ ^ = 0.

or transforming it into another whose roots are the roots of
this equation multiplied by ^q, we have

Substituting in this last equation,

z = a^y — a^x + a^, 11= ciQd^ ~ ^ii

and G = «o^<^3 — 3 a^a-^a^ + 2 a-^^

we have 2^ + 3 iT^ + (7=0.

Example. Transform the equation x^-{-2x^-\-Zx—2 = 0
into one which lacks the second term.

THEORY OF EQUATIONS 287

341. Solution of the Cubic.

z^-{-ZHz+ a = o.
Let z = ^/p + V^,

then 2-^=^ + ^4-3 ^pq ( Vp + "v/^)»

or 2^—3 Vp V^ z — Qj-{- q) = 0.

As this equation is to be identical with

z^-hSHz+ a = o,

we have, equating coefficients,

s/p -y/q z= — H^ or pq = — H^^
p-\-q= — G-.

Since we have both the sum and the product of p and q^
the quadratic equation of which they are the roots is

This quadratic is called the reducing quadratic of the
cubic. The values of p and q are therefore

p =

3-3-^ 2

Since -^p ^q = — H^

3/- H

This relation between Vjt? and ^q determines that cube
root of 5', which is to be associated with a given cube root of
p. We thus see that the apparent nine values of Vp + v g

- a

+ V6^2

+ 4^3

2

-a-

-V(7„

+ 4^3

288 COLLEGE ALGEBRA

are limited to three, as they should be to make the substitu-
tion legitimate.

For the three values of ^, we have

h = S"i + «i = ^i? 4- Vq =Vp — -^— ,

h = S"2 + «i = ^^P + co^Vq = co^p —^ = (oVp — ^^,

2!g = a^wg H- a^ = o)^^^ + wVq — co^Vp ^^-j^ = co^^p g-^,

where a^, 0^2, Wg, are the roots of the original cubic in x.
This solution, though theoretically correct, is not in a form
for practical use when all the roots are real ; it can be put
into such a form by means of trigonometry, or, better, the
solution can be obtained by the methods of 316-318.
342. From the values of 2^, z^-, %, we have

By multiplication we get

V("i - "2) ("2 - «3)('^3 - «i) = - ^^- ^^ip-q)

= _ 3 V - 3 V (^2 + 4 ^3, (yide Exs. 2 and 5, 296)
therefore

This relation furnishes a means of determining the nature
of the roots of our cubic.

If the roots of the cubic are all real, the differences are all
real, the left-hand member of the relation is positive, and
therefor<^^^2_j_4^3 jg negative.

THEORY OF EQUATIONS 289

If the roots are not all real, two of them are complex and

the three have the form

a^ = a -{■ bi,

a^ = a — M,

and therefore «j — a.^ = 2 bi,

a^ — a^ = a — c -\- b%

a^ — ci^ = a — c — bi.
Hence

<(«! - «2)'(«2 - «3)K«8- «i)'= - 4 b\<ia- cy + b^y,

and therefore (7^ + 4 JI^ is positive.

Conversely if ^ -f- 4 11^ is negative, the roots are all real,
for as we have just seen, if they were not, (t^+ 4 11^ would
be positive ; and if (^^ + 4 H^ is positive, one root is real and
two are complex, for if they were all real, G^-{- 4: H^ would
be negative.

If (7^ + 4 H^ is equal to zero, two roots are evidently
equal, and conversely.

If (r = H= 0, all three roots are equal, for in this case, the
equation in z reduces to z^ = 0, whence z^ = z^ = z^, or

S^i + ^1 — ^o^'^2 + ^^1 ~ ^0^3 + ^1 = 0, or «j = «2 = ^^3 = — — "

343. Discriminant. It may be found that (72 + 4 H^ con-
tains the factor a^. The other factor, denoted by A, is
defined as the discriminant^ that is, as the simplest rational
function of the coefficients whose vanishing expresses the
condition for equal roots. It is found that

A = ci^ci^ — Qa^a^a^^fi^ 4- 4 a^^a.^ + 4 a-^a^ — 3 a^a^.

290 COLLEGE ALGEBRA

344. The Solution of the Biquadratic. The equation

a^x^ + 4 a^ 4- 6 a^ + 4 a^ -|- a^ = 0,

(1
can by the substitution x = y — -^ (340), be transformed into

a^(^a^a^ — 4 g^g^ + 3 ^2^) — ^(a^a^ — a-^y^ _ r^
%^

or multiplying the roots of this by a^, (a; = «o^)' ^^® obtain
2!4 + 6 ^^2 + 4 (7^ + a^2j_ 3 ^2 = 0,

where /= a^a^ — 4 a^a.^ + 3 a^.

345. Assuming 2 = Vp + V^y + Vr, we have

z^ = p -\- q -{- 7' -\- 2( Vpg + Vqr + -\^rp).

Transposing and squaring again, we get
z^ — 2(p -\- q + r)^ + C?? + g + r)2= \{jpq -\- qr -\- rf)

+ %^ 'pqr(^'\/ f + V^ + Vr),
or 2* — 2 (p + 3' + r)2;2 — '^^pqr z -\- {^p -\- q -\- r)2

— 4(pg + ^r + rp) = 0.

Since this equation is to be identical with that of 344, we
have, by equating coefficients,

p + q-^r=^ —^ H,

Vpqr = -—,

Qp + q-^-rY— 4(pg' -\- qr -{- rp) = a^I— 3 H"^^
or pq-{- qr + rp=^ H^ ^ .

THEORY OF EQUATIONS 291

Hence the equation wliose roots are />, g, r, is

w3 + 3 Hii^ +Ur2_ <1\u - ^ = 0,

which is known as Eider s cubic. By adding and subtracting
the terms H^ and -*^— — , this may be written

•± ■± •±

or ^(u-\- H)^ — a^I(ii + H) + aQ^a^a^a^ + 2 a^^g^g — a^^

Putting

!<- + 11= a^v and ?/= a^a^a^ -\- 2 a^a^a^ — ^9'^ — ^0^3^ "~ ^i^^4^
we have 4 ^^^y^ — J^^^ij + e/= 0,

which is called the reducing cubic of the biquadratic. If
^1' ^2' ^3 "^^^ ^^^® roots of this equation, we have,

r = a^ — a^a^ + ^o^^3*

As ^= Vp + V^ + Vr, it might seem that there would be
eight values of 2, but as in the case of the cubic these are
limited to four by the relation

VpVo'V?' = , or ^/^ = ,

so that when jt> and q are chosen r is determined by this
relation.

292 COLLEGE ALGEBRA

If 2j, ^2, z^, z^ be the roots of the equation in ^, and ^i, a^-^ ^3,
^4 the roots of the equation in x^ we have

z^ — a^a^ + «i = V^ + Vg + Vr,

;22 = V2 + ^1 — ^i^ ~" ^S""" "^^'
2g = a^Wg 4- a^ = _ Vp + V^ — Vr,

^4 = ^0^4 + «i = — VJ9 — V^' + Vr.
STURM'S THEOREM

346. Unequal Roots. If the roots oif(x) = 0 are all differ-
ent, that is, if /(a;) and/'(:r) have no common factor and if
/2(^)'/3(^)' ">/»(^) ^^'6 functions obtained like the remain-
ders in the process for finding the highest common factor of
f^x} and/'(a:) except that in this case the sign of each re-
mainder is changed before proceeding to the next division,
we have the following theorem due to Sturm : If a and h are
any two real numbers^ b>a, the excess of the number of changes
of sign in the series of functions f{x'), f'(x')^ f^(x~), •••,/„ (2;),
when a is substituted for xover the number when b is substituted
for x^ is the number of real roots lying between a and b.

It is obvious that /,,(:r) is a constant, for otherwise /(a;)
and/'(:r) would have a common factor.

From the process for finding the functions we have the
following relations :

/(^) = ^2/2 W -/sW

/2(^)=^y3/3(^)-/4(^)

fr-l(x) = qrfXx) -fr+lix)

THEORY OF EQUATIONS 293

Here /(a;) and /'(a;) cannot vanish for the same value of x^
that is, can have no common factor, for if they had, f{x)
would contain that factor at least twice (338), but by hy-
pothesis this is not the case.

No two consecutive functions of the series can have a com-
mon factor, for if they had, it follows from the foregoing
relations that it would be a factor of all the functions of the
series, including f{x) and /'(a;), which, as we have seen, is
not the case.

To determine the loss of changes of sign in the series of
functions we have to investigate the following cases :

1. When X passes through a root of/(a;) = 0.

2. When x passes through a value which causes one or
more of the auxiliary functions f'(x)^ f^(x)^ ••• to vanish,
provided that, if more than one vanish, no two of those which
vanish are consecutive.

First Case. If a be a real root of/(rr)= 0, lying between
a and ^, then as x passes through a^fQx^ and/'(a;) have un-
like signs just before and like signs just after the passage
(339). Hence one change of sign in the series is lost. ^

Second Case. If a causes /,.(^) to vanish, that is, if
/^(a) = 0, we have/^_i(«) = — /r+i(«)- ^^ ^^^ ^'^^^ values of
X sufficiently near to a to exclude roots of /,.-i(a:) = 0, and
/r+i(^) = ^' we see, from the above relation, that for such
values /,,_i(a;) and/,.+i(a;) have signs opposite to each other,
and therefore whether /^(a;) changes sign or not as x passes
through «, the series of three functions /,._i(2:),/r(a:;),/r+i(^)
presents one permanency and one change of sign, though not
necessarily of the same order of arrangement, both before and
after x passes through «.

Thus if the signs of fr-\(x) and fr+\(x) are — and -f- re-
spectively immediately before and after, and if fr(x) is —

294 COLLEGE ALGEBRA

immediately before and + just after x passes through the

value a, we have before the passage the series of signs h ,

and just after, the series — h +, or one permanency and one
change in each case. If /^C^) does not change sign, that is,
if a occurs an even number of times as a root of fr(^x) = 0,
we might have, for example, the series — h + both before
and after the passage. Hence in the second case no change
of sign is lost or gained in the series of three functions. If
others of the auxiliary functions vanish, the same thing is true
for each of the corresponding series of three functions.

Thus we have proved that as x passes through a real root
oif(x) = 0 one change of sign is lost in the series of func-
tions/(a;), /'(a;), • ••,/„ (a;) and in no other case is a change of
sign lost or gained. Hence the number of changes of sign
lost as X passes from a to b is the number of real roots of /(a;)
= 0, lying between a and h.

The loss of changes of sign between /(:r) and /' (a;) hap-
pens by means of the rearrangement of the signs of the
series as x passes from root to root.

347. Equal Roots. If a occurs r times as a root oi f(x) =
0, that is, if (re — ccy is a factor of /(r?^), {x — a)'-i is a factor
oif'(x) (338), and by the highest common factor theory, it
will be a factor of each of the functions /g (a;), /g (a;), ■'■^fj^(x)^
where /^.(a:) is that remainder which exactly divides the pre-
ceding one and with which the process terminates. Simi-
larly, if f(x) contains other multiple roots, fk(^x) contains
them each to a degree one less than they are contained in
f{x)* Consequently, if we divide the functions fix)-,
f'(x)^'"^fj^(x) and also the relations existing between them
(346) by fk(x)^ we obtain a new series of functions F(x)^
F^(x)^ •••, Fj^(x\ where F(x) contains each distinct factor of
f(x)^ once and only once, and a new set of relations. It is

J

THEORY OF EQUATIONS 295

apparent that the terms in this series of functions will have
the same signs or signs opposite to those of the original series
according as/;-.(^) i^ positive or negative, and hence v^ill pre-
sent the same changes or permanencies as the original series.
Therefore we have a new series of functions possessing the
same properties with respect to loss or gain of changes of
sign as the functions of 346, and hence the reasoning and
results of that section are applicable to the function F(x). .
Hence Sturm's theorem holds for real multiple roots, count-/
ing each multiple root once.

Example. Determine the number and situation of the
real roots of the equation 2^ — 3 a;^ + 5 a; — 1 = 0.

The signs of the terms are -\ 1 , whence by Descartes's

rule there cannot be more than three positive roots. Chang-
ing X into — ic, the signs are all negative, hence there can be
no negative root. There is at least one positive root, since
the absolute term is negative. To determine whether all
the roots are positive we must use Sturm's theorem. The
work is as follows:

f(x) = x^ — ^ x^ + b X — \^
/(a;) = 3 2:2 -6 a; +5.

Dividing 3/(a;) by/^(a;), we get 2: — 1 as quotient and 4:x-\-2
as a remainder. Dividing this remainder by 2 and chang-
ing its signs, we have —2 2; — 1 for/2(2:). Dividing 2/' (2;)
by f^Cx), we get — 3 2: + 15 as quotient and 35 as a remain-
der, and therefore /3(2:) = — 35.

For x= — OD the signs of the functions are h H — and

for x= ao they are + H , therefore as one change of sign

is lost there is only one real root. For 2: = 0 the signs are

— I and for x=l they are + H , and as one change

of siofn is lost between 0 and 1 the root lies in that interval.

296 COLLEGE ALGEBRA

348. Solution of Numerical Equations. The real roots of
numerical equations are either commensurable or incommen-
surable, and their number and situation may be determined
by the use of Sturm's theorem and the principles of 319-325.
The former class includes integers and rational fractions,
and since every equation can be reduced to one having the
coefficient of the first term unity, and the coefficients of its
other terms integers, the problem of finding the rational
roots of an equation reduces itself to the problem of finding
the rational roots of an equation of this type.

Let the equation of this form be

x'^ + a^a;^-! + a^x""-^ H h «« = 0

where the a's are integers. The commensurable roots of this
equation must be integers, for, if not, suppose that ^, a com-

mensurable fraction reduced to its lowest terms, is a root.
Then

or multiplying both members by q^~'^ and transposing,
— — = <2jjt?"i + a^p^ ^q-{- h a,,q^~'^'

The right-hand member of this last equation is an integer,
while the left-hand member is not an integer, since p and q
are prime to each other, hence the supposition that our equa-
tion can have a fractional root is false, and hence its com-
mensurable roots must be integers.

Since a„ is numerically the product of all the roots, we
may find the commensurable roots by using the integral
factors of a^ according to the method of 333.

THEORY OF EQUATIONS 297

Example. Find the commensurable roots of 4 a:* — 8 o:^
4- 23 x^ — 4:0 x+ 15 = 0. Forming an equation whose roots are
the roots of this equation multiplied by 2, we have (t/ = 2 x^

y - 4 ?/3 + 23 ?/2 - 80 y + 60 = 0.
The factors to be tried are ±1, ±2, etc. Dividing by ?/ — 1,
we have 1 _4 23-80 60

1 -3 +20 _60
1-3 20-60

which shows that 1 is a root, and i/^ — S ^^ -^ 20 t/ — QO is the
quotient. If we divide the quotient by ^ — 2, we have

1 _3 20-60

2-2 36

1 _1 18-21

which shows that 2 is not a root. Similarly we may find
that 3 is a root, and that the other factors are not roots.
Since 1 and 3 are roots of the equation in j/, ^ and | are the
corresponding roots of the equation in x, and are its only
commensurable roots.

349. Horner's Method. When it is found that a positive
root lies between two consecutive integers, we may form an
equation whose roots are the roots of the given equation
diminished by the smaller of the two integers which is the
integral portion of the root sought. If now we multiply the
roots of the resulting equation by ten, we can again find two
integers between zero and ten such that a root of this equa-
tion lies between them. This root divided by ten is the
remaining portion of the original root, and therefore the
smaller of the two integers is the next digit of the root. By
a repetition of this process as many of the digits of the root
may be obtained as is desired.

298 COLLEGE ALGEBRA

Thus given f{x) = 2^ -I'lx^ +?Ax - b = 0,

We find tliat/(3) is positive while /(4) is negative, indicat-
ing the presence of at least one real root lying between 3
and 4. Sturm's functions are as follows :

f(x) = 2^3 _ 12 ^2 ^ 31 ^ _ 5^
/(^)=3a;2-24 2:+31,

It is not necessary to find f^^x^ since all we need is its sign,
which can be determined without finding the function. The
sign of this function is positive, since the value Jg^- for which
/2(^) vanishes makes/'(a^) negative, and therefore y3(a7) must
be positive (346, 2).

For 2:= — 00, tlie series of functions have the signs — | h,

and for a; = 00, the signs + + + + • Hence between — 00 and
-{-co three changes of sign have been lost, therefore the
equation has three real roots. Again for x = 3, the series of

functions have the signs H h and for :c = 4, the signs

are f- +, that is, one change lias been lost, and therefore

but one root lies between 3 and 4. The work of obtaining
the rest of the root to four decimals is as follows :

Diminishing the roots by 3

1 -12 31 -5

3 - 27 12

1

-9
3

4 T

-18

1
1

-6

3

-3

-14
Multiplying the roots by 10 and
then diminishing them by 4

THEORY OF EQUATIONS

299

1

-30
4

-1400
-104

7000
-6016

1

-26
4

- 1504

-88

984

1
1

-22

4

-18

- 1592
Multiplying th
climinishin

le roots by 10 and
Lg them by 6

1

-180
6

- 159200
-1044

984000
- 961464

1

-174

6

- 160244
-1008

22536

1

1

-168

6

-162

-161252
Multiplying the roots by 10 and
diminishing them by 1

1

-1620
1

- 16125200
-1619

22536000
-16126819

1

-1619
1

-16126819
-1618

6409181

1
1

-1618

1

- 1617

-16128437
Multiplying the r
diminishing

oots by 10 and
them by 3

1

-16170
3

-1612843700

- 48501 -

6409181000

- 4838676603

1

- 16167
3

- 1612892201
- 48492

1570504397

1

- 16164

- 1612940693

1
1

3
-16161

1

- 16161

- 1612940693

1570504397

300 COLLEGE ALGEBRA

The root to four decimal places is therefore 3.4613. This
method of obtaining a root of an equation is known as
Horner's method.

350. If two roots are nearly equal, that is, if the digits are
the same in each to some point in their decimals, we proceed
as before until Ave come to a point where two roots are found
to lie between two different consecutive integers between zero
and ten, when they will begin to separate and each can be cal-
culated separately. The complete solution of this case, how-
ever, requires more detail than the scope of this work admits.

If a root is negative, we may transform the equation into
one whose roots are the negatives of those of the given equa-
tion and proceed as for a positive root.

351. EXAMPLES
Solve the equations:

1. x^-6x'^-hllx-6 = 0, 4. x^-1x-6 = 0.

2. x^-2x'^-x-\-2==0, 5. x^-^2x^-5x~6 = 0.

3. x^-2x^-5x-^Q = 0. 6. x^-7x-{-Q = 0.

7. x^-5x^-(j4:x-i-U0 = 0.

8. x^-103^-\-S5x^-50x-{-24: = 0.

9. x^-7x^-^bx^-\-Slx-S0 = 0.

10. a:4-19a;2 + lla; + 30-0.

11. x^-7x^-\-dx'^-7x-10 = 0.

12. x^-Ux^-^4:9x^-S6 = 0.

Suggestion. Put x^ = z.

13. 2:6 — 52:^—2:2+5=0. 14. 2:^-2:6-64 2:3+64 = 0.

15. G,T3_8ia,2_p53^_30=0.

Suggestion. Transform to an equation having the coefficient of the
highest power of x equal to unity.

THEORY OF EQUATIONS 301

16. S0a^-llx^ + 59x-U = 0.

17. iJx^-lr^-lS3^-{-Ux-h6 = 0.

18. x^ — x^ — 9 x'^ — 3 X + 2 = 0.

19. 12 x^ + 44 x^ + 23 2;2 - 28 2: + 5 = 0.

20. lSx^ + S'^x^-2x^-7 = 0.

21. x^-7 x-]-o = 0.

Suggestion. Apply Sturm's theorem and plot the curve. Solve by
Horner's Method.

22. a:3 + 5a; + 3 = 0.

23. Find the positive root of 2:^ — 6 :r — 13 = 0.

24. Solve the equation x^ + x^ -{- 1 = 0.
Suggestion.

x<i^x^+l=:0 or x^ + l-^—=x^ + s(x + -\ + — -3(x-\--] + l

x^ V x) x^ \ x)

= {x + -y -?,lx + ~\ + l=z^-^z+\=0.

25. Find all the roots of the cubic

and show that the equation ma}^ be regarded as a reduced
form of x^ + x^ -\- x^ -\- x^ -{■ x"^ -\- x + 1 = 0.

CHAPTER XIX
MISCELLANEOUS TOPICS

MATHEMATICAL INDUCTION

352. Some further work on mathematical induction is here
given for those who care for it. In 169 mathematical induc-
tion has already been used to prove the binomial theorem.

353. Divisibility of x'^ ±y^ by x ±y. We know by trial

that x^ — y^^ x^ — y^^ are divisible by x — y.

Since x^ — y'^ = x^ — xy'^~'^ + xy^~'^ — y^

= x(x''-^ - /^-i) + y^'-^ix - ?/),

we see that \i x — y divides a;""i — ?/^~i it also divides x^ — y^^
for any value of n — 1. We have seen, however, that a;^ — y^
is divisible by x — y^ therefore x^ — y^ is divisible by x — y^
hence x" — ?/^, and so on continually. Hence x"^ — y'^ is divis-
ible by X — y^n being any positive integer, and x and y hav-
ing any values. The quotient is

^ = x""-^ -I- x^'-'^y + x^'-^y'^ H h 2:2/^-2 + ?/"-i .

^~y (See Ex. 10, 86)

If in this identity we change y into — y, we have

^!^l! = a;'^-l _ ^^-2^ ^ x^'-^yi + ... _|_ (-l)'^-2.^:^»-2

according as n is even or odd.

Since x"^ + ^'^ = 2:" — ^" + 2 ^^, x"^ + y'^ is not divisible by
x-y.

302

MISCELLANEOUS TOPICS 303

Thus for a positive integer w,

x—y never divides a;" -f ?/",

X — y always divides ^" — ?/%

x-\- y divides x^ — y''^ when n is even,

x-\- y divides x^ + y'^ when n is odd.

The student may prove that

x-\- y does divide x^ — y^ when n is odd, and that x+ y
does not divide x'^ + ?/^* when n is even.

354. Summation of Series. The method of mathematical
induction may sometimes be applied to find the sum of a
series.

EXAMPLES

1. Find the sum of the series 1, 3, 5, etc.

The sum for one term is 1, for two terms is 4, for tliree
terms is 9, that is, „ _ i „ — 92 « _ q2

*1 — ^1 ^2 — ■" ' *3 — ^ *

Let n be the greatest value for which and for all lower
values of which we know the law Sn = n^ to be true. Then
since ^^^ = 2n-l, if«+i = 2w + l,

s,,^^ = s„ + W;,+i = w2 + 2 ^1 + 1 = (?^ + 1)^-
Hence the law holds for a value one more, and therefore

for any positive integral value of n.

2. Prove that the sum of the series

1.2 + 2-3 + 3.4+ ••• +</i + l) is l?iO^ + l)(w + 2).

We are to prove here that

s„ = iwOi + l)(n + 2)

304 COLLEGE ALGEBRA

The reasoning can be carried out as before, or it may be
obtained by indirect reasoning as follows :

si = 1.2 = i(l)(2)(3),

«2 = K2)(3)(4),

S8 = K3)(4)(5).

If the law here observed fails, it must fail for a first time,
say for the (ti + l)th case. Then by hypothesis it does not
fail for the nth, and hence

and therefore

««+i = Sn + ^M+i = i n(n + V)(n -f- 2) + (?i + V)(n 4- 2)

= (n + 1)(^ + 2)(i 7i + 1) = IQn + l)(n + 2)(n + 3),

hence the law does in fact hold for the (n + 1) th case also,
and no first failure does occur, and

s„ = i?i(w + l)(n + 2)
is true generally.

355. Prove

> (^i«i + A^a^ + ••• + A^a,^'^.

Let n be the greatest value for which and for all smaller values
this inequality has been shown to be true. Then, since

A If, 2 _ /| If, 2

J 2^2i/12^ 2-^<9J^J ^
-^^n+i ^^2 ' 2 "'w+1 ^ "^ -'-*-2"'2 ^i+l^i+1'

>4 2^2_L./12^ 2-^9/1/yJ /7

(^A^^A;--^ ... +A„2)(a^2_^^^^2+ ... +^^2>)

> iA^a^ + ^2^ + ••• ^«^«)^'

MISCELLANEOUS TOPICS 305

we have, adding corresponding members of the equation and
of the inequalities, and combining,

Hence the inequality is true for a value of n one greater.
But when n is 2 it is easily seen to be true, and therefore it
is true when n is 3, and when n is 4, and so on generally for
any positive integral value of n.

356. EXAMPLES

Prove by mathematical induction :
1. 1 . 3 .5 ••• 2 7i-l<n'\

nJ \ nJ \ n J n\

3. 1.3-f2-4 + 3-5+---to7i terms

= J^i(7i + l)(2n + 7).

8. 13 + 03 + 33 + ... + n^ = Mn±llJ.

LIMITS

357. In 190 we have stated four elementary theorems of
limits. Before we can satisfactorily prove them it is neces-
sary to consider briefly the theory of indefinitely small and
indefinitely great numbers.*

358. Definitions. An indefinitely small number is a variable
whose absolute value under the conditions of its statement may
be made less than any assigned positive number however small,

* These are the infinitesimals and infinites of calculus.

X

306 COLLEGE ALGEBRA

Hence we see that its limit is zero, but the indefinitely small
number itself is not zero.

An indefinitely great numher is a variable ^vJiose absolute
value under the conditions of its statemefit may be made greater
than any assigned positive number however great. It follows
that it increases without limit, but is not itself equal to
absolute infinity, i.e. to the reciprocal of absolute zero.

359. Theorem. The reciprocal of an indefinitely small number
is indefinitely great., and the reciprocal of an indefinitely great
number is indefinitely small.

Proof. Let x be any indefinitely small number and let A
be any assigned positive number. Then — is also a positive

r^ 1

number that can be assiscned. Therefore x< — , therefore

1 1 . . . ^

-> A^ therefore - is indefinitely great.

XX

Let X be any indefinitely great number. Then X>— ,

11

therefore — <A, therefore — is an indefinitely small number.

X X -^

360. Theorem. The product of an indefinitely small number
by any number a not indefinitely great is indefinitely small.

Proof. Using the notation of 359, we can so assign A that

1 1
a<A^ and therefore -> — . Let a be assigned, then

11 " ^ . .

2: <—«<-«, and ax<a^ or ax is indefinitely small.
A a

Corollary 1. The quotient of an indefinitely small number
hy any number not indefinitely small is indefinitely small, for

if a = — , the preceding result is —. <a.

a a

Corollary 2. The product or quotient of an indefinitely
great number by any number not indefinitely small or great

MISCELLANEOUS TOPICS 307

respectively is indefinitely great, for from a; < — « < - «, it

A a

1 A a 1

follows that -> — >-, or as we may write it, since - = JT,
X a a X

-X'>-, — >-, a'Jr>-, where a is not indefinitely great.
a a a a

361. Theorem. Tlie product of an indefinitely small num-
ber and an indejinitely great number may have any value.

Proof. If a is not indefinitely great, — is an indefinitely

^ ^ ax

great number by 360, and x x ^ = -, which may have any

ax a

value not indefinitely small. Again if a is not indefinitely

small, then -—. is some indefinitely small number b}^ 359

11

and 360, Corollary 1, and X x —— = - , which may have any

aX a

value not indefinitely great. Hence the product may have

any value.

Corollary. The quotieyit of tiuo indefinitely small or of two
indefinitely great numbers may have any value.

362. From the preceding theorems w^e may tabulate the
following results, x and y being two indefinitely small
numbers, and X and Y two indefinitely great numbers.

xy is indefinitely small. XY is indefinitely great.

a:— is indefinitely small. JT- is indefinitely great.

X x

xX\^ indeterminate. yZ is indeterminate.

— is indeterminate.

y Y

X

X X

— is indeterminate. — is indeterminate.

/ Y

X . . . X

-— is indefinitely small. — is indefinitely great.

308 COLLEGE ALGEBRA

363. Theorem. The sum of n indefinitely small yiumhers of
the same sign is indefinitely small provided n is not indefiriitely
great.

Proof. Let e^, e^^ •••, e^ be ^ indefinitely small numbers;

and let a be an assigned positive number ; - is also an as-
signed positive number. And

n

I I ^ ^

^2 < - ■>

n

\^n\<-'

n
Therefore I ^i + ^2 + *" +^n|<a» (26)

and hence ^1 + ^2 + " " ' + ^«

is indefinitely small. But if n be indefinitely great, the in-

equality i— ^ < 1, which before held, and conditioned the con-
a

n
elusion, does not now necessarily hold, for by 361, Corollary,

^-^ may have any value, and the conclusion no longer neces-

n

sarily holds. This theorem is of the utmost importance.

Corollary. The sum of a finite number of indefinitely small
numbers of different sig7is is indefiiiitely small or else zero.

364. Theorem. The sum of any number of indefinitely
great numbers of the same sign is indefinitely great; but if the
numbers have different signs, the sum is iiideterminate.

MISCELLANEOUS TOPICS 309

Proof. In the first case X^, X^, •••, X„, being indefinitely
great,

A

and since I Xj. I > — ,

therefore, | Xj | + | Xg | + • • • + | X„ | > J..

For the second case X may be any positive indefinitely
great nnmber, and a any positive number whatever. Then
X>A, X+a:>A, and X-{-a=Y is indefinitely great,
Y—X=a, X— Y= — a, and the difference between two
infinitely great numbers of the same sign may have any
value whatever. Another proof of this might be given as
follows: -, -,

y X xy
which by 363, 360, and 361, Cor., is indeterminate.

365. Theorem. The difference hetiveen a variable x and its
limit a is an indefinitely small number. For by definition
188 a — X can be made as small as we please, that is, \a — x\<a^
where l>a>0, ov x — a — e, x= a + e, where e is indefinitely
small. We note that equivalent statements are :

x = a^ x—a = €^ x=a-{-€^ a-{-a>x>a — a, a>x—a> — «, Lx=a.

366. Theorem. If tivo variables are constantly equal and
each api^roaches a limits their limits are equal.

Proof. Let x and y be two variables, a and b their
limits. Then we have x= a-\- a, y = b -\- ^, wdiere a and fi
are indefinitely small and x = y.

Therefore a _ J = /3 - «.

310 COLLEGE ALGEBRA

Now y8 — « is either indefinitely small or else zero by 363,
Cor. It cannot be indefinitely small, for a ~h is not a
variable, therefore it must be zero; therefore a = h.

367. Theorem. The limit of the algebraic sum of a finite
number of variables is equal to the algebraic sum of their limits.

Proof. Let 2^^ = a^ + e^, x^ = a^-\- e^, • • •, x^ = a^ + e^,
where a?-^, x^-, •••, x-n, are n variables, ^j, a^, •••, a^, their limits,
and e^, 63, •••, e„, n indefinitely small numbers.

Then

x^-\rx^-}- • • . -{-x^= a^ + a^-jr • • • + a„ +€-^^-^e^-\- • • • + e^.

But by 363, Cor., €-^ + e^-\- ••• + e,^ is either indefinitely
small or else zero when n is finite. Therefore by defini-
tion ^j 4- «2 + ••• +^rt is the limit of x-^-}-x^+ ••• + x^^ and

hence

IjZx = 2a = 2.Lx.

368. Theorem. The liynit of the product of a finite number
of variables is equal to the product of their limits.

Proof. Using the notation of 367,

= .,a,.-a„(l+^)(l + ^

a„

M^-J tto * * * ttj

= a-ittn ••' a

\ ^"^ a-^ '^ a-^a^ "^ a-^ • • • t«„

\ a-i tt -1 • • • tfjji

(326)

MISCELLANEOUS TOPICS 311

By 360 and 363 the expression in the parenthesis is, wlien
n is finite, indefinitely small unless zero, and hence by 360
the second term of the right-haud member is indefinitely
small or else zero, and hence by definition a-^a^ ••• <^n is the
limit of

X

^x^ ••' Xfi^ or Lx-^x^ •■■ x,i= a^a^ • • • (in= Lx^Lx^ • ■ • Lx,^.

369. Theorem. The limit of the quotient of tivo variables is
equal to the quotient of their limits.

Pkoof. With the previous notation

x^ a^ -h ej

x^ a^ a^-\- €^ a^ a^^i — a-^^
x^ a^~ a^-\-e.^ a^ ^2(^2 + ^2)

The denominator a^ (a^ + ^2) is not indefinitely small as
long as a^^Q^ but the numerator 6^2^^ ~ <^i^2 ^^ indefinitely
small by 360 and 363, Cor., unless it is zero; hence

Z -^ = -i, by 365, or X-i = y-^-

Corollary. If two variables are in a constant ratio, their
limits are in the same constant ratio, for

XOC"{ Ct-t

It will be noticed that in this corollary, as well as in other
places, it is convenient to speak of the limit of a constant as
that constant.

312 COLLEGE ALGEBRA

370. EXAMPLES

1. Prove that the limit of a power of a variable is equal
to that power of the limit of the variable when the exponent
is any finite real constant.

2. The limit of the root of a variable is the root of the
limit of the variable.

ON THE CONVERGENCE AND DIVERGENCE OF SOME
PARTICULAR SERIES

371. In what follows generalizations with respect to the
convergence or divergence of a power series

I ^Q I + I u^x I + I u^x^ I + • • • + I u„x^^ I + • • •

of positive terms, where ^f„ is an algebraic function of n, and
related problems will be considered.

372. 1. Definition. In the elementary sense an algebraic
function of a variable is 07ie wJiieh is obtained by performing a
finite number of operations of addition^ subtraction^ multiplica-
tion^ division^ involutio7i, and evolution on the variable. In the
higher sense of the theory of functions^ x is an algebraic f mic-
tion of 9^, tvhen x ayid n are connected by an algebraic equation
F(x^ n) = 0. It is seen that the second sort of function
includes the former as a special case.

2. Definition. The total degree of an elementary algebraic
functio7i is, as has already been stated in 210, the degree of
the numerator miiius the degree of the de7iominator, e.g.,

(n -\- a')(n-\- b)(n -{- <?)

n{n + 1)(^ + 2)(n + S)(n + 4)
is of total degree — 2.

MISCELLANEOUS TOPICS 313

V?i — 1

1-

is of total degree l — | = — 1.

The total degree of a higher algebraic function 61 the sec-
ond sort is defined as follows : If p is the value which makes

where c is finite and not equal to zero^ then — p is the total
degree of x=f(ji)^ obtained from the relation F(x^ n) = 0.
It is seen that this definition may be regarded as the gener-
alization of the former and includes it. (See 373, 2.)

373. LIMITS OF RATIOS

1. If f(n) is an algebraic function of either sort of total de-
gree zero^ its limit as 7i = oo is finite and not zero. When
numerator and denominator are developed, we have for a
function of the first sort, t^ ^

^. . _ An' 4- Bn'-'^ + On'-* + .-• _ n'' if

-^^"^^ ~ A'n' + B'n'-'' + C'n'-" + -'~ ^> j^B[ C[ '

n" n"

whence, L fQii) =—j = c,

n = 3o J±

where c is finite and not zero.

If /(n) is a function of the second sort, this proposition is
true by definition.

2. If /(w) is an algebraic function of the first sort, we
have the following theorem : That 2vhich is taken as defini-
tion of total degree for a function of the second sort is true as
a property of a function of the first sort., and conversely.

Proof, hetf(n') of the first kind be of total degree — jt?,

314 COLLEGE ALGEBRA

tlieii/(^)^^ is of degree zero, hence its limit, as ?^ =00, is by
1 finite and not zero. Conversely, \if(n)n^ has a limit which
is finite and not zero, /(92) must be of total degree —p^ as
defined for a function of the first kind, for the assumption
that it is of different degree leads to a limit 0, or cx). This
identifies the definition given for the total degree of an
elementary algebraic function in 372, 2, as a special case of
that given for one of a higher kind.

8. If f{n) is a function of either sort^ then

L(mod) /^"^^ =1.

n = oo f(n — 1)

For, first, since the total degree of ^-^^ ^. is zero, the limit

f(n - 1)

of the ratio is finite and not zero. Second, if — jt> be the total

degree of f(n) of the first sort, f(ii)iRP is of degree zero and

by 2 its limit, as 9^ = 00, is c. Similarly, since

L f(n-V)n^= L fQii-V)Qi-iy = ±c.

« = 00 W — 1 = ao

Hence
L (mod) -^^''^ = L (mod) /(^^ = (mod) ± - = 1.

If /(ti) is of the second kind,

L (mod)/(9^)w^,

• n = <x>

L (mod)/(w - l}(n - 1)^

n — 1 = 00

L (mod-) f(n - l)(n - 1)^

«=oo

and L (mod) f(^ti — 1) n^

MISCELLANEOUS TOPICS 315

are identical, the first three obviously, and the fourth is
equal to the third because

4. We also note the theorem :

L • J^ = 0, or oo,

according as p^^p^-, tvhere —pi and —p^ ^^^ ^he total degrees
off I andf^.

374. THEOREMS ON CONVERGENCE

1. All positive series ivhose nth terms, u„^ t?„, tv^, •••, a?'e alge-
braic functions of n of the same total degree are convergent
together or divergent together. For let

be the nth terms of any two of the series. Then since

= c.

L /lOO

where c is finite and not zero (by 373, 1), the hypothesis in
207 is satisfied, and therefore the conclusion follows. By
means of this theorem we may classify series of the given
kind with respect to the total degrees of their nth terms,

and since the series whose nth term is v,^ = ^ of total degree

n^

—p, is convergent if j9 >1, but otherwise divergent, we have
the following theorem:

2. If the nth term of a positive series Un =fO^^ *'^ ^^ alge-
braic function of total degree —p-, the series is convergent if
jp > 1, otheriuise it is divergent. Thus the determination of
convergence or divergence becomes simply a matter of deter-

316 COLLEGE ALGEBRA

mining the total degree of u^^ and this can often be done by
mere inspection.

3. If the nth term of a positive series he Uj^=f(ri)x^~'^^
where f is an algebraic function of n^ then the series is conver-
gent if x<l^ divergent if x>l, and when x=l^ convergent if
p>l, otherwise divergent^ where — p is the total degree of
f(n). The proof of this theorem follows from the ratio
test (205, 206). According as

I . -^ > 1

«iOO lir

<

the series is convergent or divergent. When

and

L ^" =1,

a further test

is

required.

If

J.,

or if

and

L '^'^ -1,

n^ca 11,

the series is divergent. We have here

L — —= L /^ ^ x = x

n=(Xi 'If'n-l n=ccj(^n LJ

(by 373, 3), and we have in fact, when x<l the series is con-
vergent, when x>l the series is divergent, and when x=l
the series is convergent or divergent, according as^^l.

MISCELLANEOUS TOPICS 317

4. In case of a series of positive and negative terms or in
case of a series with complex terms, we may consider the
series i i • i i ■ i 9 1 ■ ■ 1 « 1 ■

I Wq I + I U^X I + I U2X'^ I + ■ • • + I '^n^ I + '"■>

and the well known theorem (214), if the modular series of
a given series is convergent, the series is also, leads to other
useful applications. From what has already been said it is
not necessary to supply further details for the treatment of
this case.

It is to be carefully noted that in the foregoing, factorial
functions, exponential and transcendental functions, and all
forms of higher functions have not been considered.

375. EXAMPLES

1. The series

(l + a)(l + ^)(l + 0 , (2 + ^)(2+^)(2 + 0
1.2.3.4.5 2.3.4.5-6

■^ nQi + l)Oi + 2){n + 3)(n + 4)

is seen by inspection to be convergent if x<l, divergent if
a; > 1, and when x = 1^ convergent, since the total degree of
f(n} is - 2.

2. The series

12 71

1+V2 I + 2V3 1+nVn + l

is seen at once to be divergent since the total degree of /(w) is

1 _ 3 = _ 1

2 — 2*

318 COLLEGE ALGEBRA

3. The series

is seen in the same way to be convergent if a: < 1, diver-
gent if x>l^ and when x=l, divergent (total degree of
f(n) = -l; see 372).

4. The series whose nth term is

Un = n''( Vw — 1 — 2 Vw — 2 + Vn — o)x''\

is convergent when x<l^ divergent when x>l^ and when
x= 1^ 'd little calculation shows that p will be > 1, if /c < J ;
hence Avhen rr = 1, it is convergent if a: < ^, otherwise diver-
gent.

5. The series whose nth term

where /(ti) is obtained from the cubic equation

a^a^ + (^0^ + h-^)x^ + (Cq7i^ + c-^7i -\-C2)x + d^n^ + d-^n'^ -f- (^g^ + f/g = 0,

is convergent if 2 < 1, the modular series is divergent if 2; > 1,
and when 2; = 1, the modular series is divergent, since

f(Yi) = X and = GO , as 7^ = 00 .

To show this, put n = — \

m

the equation then becomes:
a^mV + (h^ifr? + h^)x^ + {c^m + c^irp- -f- c^n^~)x + t7Q+ (i^m

+ c?2^^^ + d^ — 0,
when in fact if m — 0,

all three roots become infinite.

MISCELLANEOUS TOPICS 319

It ^0 ^^ 0 ^^ ^0 ^^ 0'

the equation approaches the form

Qmx)^ + (^mxy^ + (^mx) + 1 = 0.
i.e. mx = a^

where a is a complex fourth root of unity,

x = an,
hence /(w) is of total degree 1.

6. Given F{x, n) = 7i^x^ + 4 nx"^ -f- 3 = 0,

or a;^ + 4 mV -\-om^ = 0;

as m = 0, a; = 0,

Lff£Y + 4f^):. + 8)=0,
m^\\mj \mJ J

or

L

. 3^0)
La: = ^,

where 0)^= 1,

and the total degree of /(w) is - 1. Hence the series whose
nth term is \un\ = \f{n) \ is divergent.

7. F{x, 7^) = n^x^ + 2 nH^ + 2 = 0,

or a;3 + 2 ?7i%2 + 2 7?z6 = 0,

as m = 0, a:= 0.

which shows that x is of total degree - 2. Hence the series
defined by u^ =f(ji) = x\^ convergent.

320 COLLEGE ALGEBRA

8. F(x, n) = n^x^ + 3 nV + 5 = 0.

(^Y + 3m-3(4y + 5 = 0.

x=f(n) is of total degree — |, and the series whose nth.
term is 2/;^ =f(n)z"~'^ is convergent if 2; ^ 1.

9. Let the student work the problems of 215.

376. The Product of Two Infinite Series.
Theorem. If two infinite series

JJ= Uq-\- u-^x + v^x^-]- h Uj^x"^ + • • •

and V= Vq + v^x + v^x'^ + • • • + v„a;"+ • • •

are absolutely convergent^ then the third infinite series

P = u^v^ + (u^v^ + u-^i^Q^x + (uqV^ + 11-^1^ + ^2^0) -^"^ H 1" (V«

+ UiVn_i H h ^^,^'^0)2;" + • • •

in which the coefficient of any po^ver of x is the same as in the
product of U and V, is also absolutely convergent and equal
to UK

We only need to prove the theorem for the modular series
or series composed of the absolute values of the terms of the
given series, since a series is convergent if its modular series is
convergent, by 214. Denoting the modular series by accents,
we have ZZ'ga X V'^n = ^'2'i + terms containing 2^" and higher
powers of x where U'^w V' ^n^ P\n denote the sums of the first
2 n terms of the modular series of u, v, p respectively ; P' 2,n
= U'n X V'n + other terms. Hence U' ^,, V'^^ > P'^„ > U\ F'„;
taking the limits of the three expressions in this inequality,
and noting that L U\j, = L U\, = U\ L W^j, V\n = ^' y\ etc.,
we see that LP' ^^ = LP' ,, = P' = U'V. '^ Thus L(iU\,,V' ^^
— P^ji) = 0, and since U'^nV'^n- P'2n^ U^^nV^n- P^^n^ it fol-
lows that X( U^,, V^„- P2n) = 0, or P = UV.

MISCELLANEOUS TOPICS 321

377. Vandermonde's Theorem. If r, s, and n be positive
integers such that r-{-s>n. we have proved in 162 that

r+s

^n — r^n~^r^n-l ' s^l + r ^n-2 * •» ^2 "^ " * "^ 5^»*

Multiplying each member of this equation hj nl^ we have

''^A T^ 7-i . /^n\ 7-» 7-» , , /^

Denoting „P^ by n^, this result takes a more striking form :
a formula which could be obtained from

?%

by changing all the exponents into subscripts. Since (2)
is of degree n in r or s, and since it is true for more than n
values of r and a value of 8 for which r + s > ?^, it is true
for all values of r, and in the same way it can be shown to
be true for all values of s and a value of r by 83. Hence
it is true for all values of r and for all values of s. The
identity

(r + s)„ = r,, + (jj r„_iSi + Q) ^»-2«2 + " * + ( J ^»
is known as Vandermonde s TJieorem.

378. The Binomial Theorem for any Index. We give here
another proof of the binomial theorem, see 221.

Let

/(^)^l + ^^+|^2^2+... + ^r^r^_.... (1)

322 COLLEGE ALGEBRA

Then /(^)=l + ^l^ + ^^2_^...^^r^r_^...^ (2)

and

fQm+n)~l + ^ ^^x-{-^ ^2a:^H \-^ — y^a:^+ •••(3)

J. I ^ I T I

f(m) and /(9i)are absolutely convergent series when | a: | < 1
(221), and the coefficient of x^ in/(m) xf(n} is

7-! (/•- 1)! 1! (r- 2)! 2! (r-s)lsl rV

If . . r!

r I V (r — s) ! s I /

which by Vandermonde's Theorem is equal to ^^ — — — ^. (4)

Thus the coefficients of the different powers of x in f(jri)
xf(n) are always equal to the coefficients of the correspond-
ing powers of x in f(rn + ti), and therefore by 376

/(m) xf(n) =f(m -}- n), (5)

provided | a^ | < 1.

By using (5) repeatedly we obtain

/(m) xf(n) xfQp) x •.•=/(m + n + ^ + •••)• (^)

Let m — n—'p— ••.=-, r and s being positive integers, and

s

there being s of the numbers w, ?2, jo, •••, and (6) becomes

[•

^e

=/?:x. =/(r). (7)

But when r is a positive integer, /(r) = (1 + xY'

MISCELLANEOUS TOPICS

323

Therefore

and

•'li

= 0- + xy,

(i + .)^=/g

(8)

(9)

This proves the binomial theorem for a positive fractional
commensurable exponent. By limits as in 221 the proof
can be extended to any positive exponent.

Again /(0) = 1, and by (6)/(-n) x/(n)=/(0) = l. (10)

Therefore /(— 7i) =

Y

A

f. . = 7T^^ = a + ^)-"by(9)and91.
f(n) (1 + xy

Hence (1 + a;) "»=/(- w), (11)

which proves the theorem for any negative exponent. Hence
the theorem is true for any real exponent provided | a:|<l.

379. The Complex Variable as Function of its Modulus and
Argument. We have seen in 126, 129 that the complex variable
z = X -\- yi may be
represented by the
point P = (x, y) in
the xy plane ; that
to every value of z
there corresponds a
single point P =

(a;, ^), and con-

versely to every

point P = (a:, y')

there corresponds a

single value of 2,

z = x-\- yi^ and that

z may be expressed as 2; = r(cos <\> -{- i sin c^), where, Fig. 34,

r = V2;2 + ^/2 is the modulus of z, and ^ is its argument

or amplitude.

o

X

Fig. 34.

324 COLLEGE ALGEBRA

When the modulus of a complex variable vanishes^ the variable
vanishes and conversely. Yov if 2; = 0, r(cos <^-\- i sin <^) = 0,
and either r = 0, or cos </> + ^ sin (/> = 0 ; but it is impossible
for cos (^ + ^ sin <^ to vanish, for cos (/> and sin <^ do not vanish
together ; therefore the vanishing of z is due to the van-
ishing of r alone. This is seen geometrically in that z=0
represents the point 0, or the origin, and only for the point
0, or the origin, does z vanish.

Similarly a complex variable z is infinite when and only
when its modulus is infinite.

380. De Moivre's Theorem for a Positive Integral Index.

We shall prove here that for a positive integer m, (cos cc -{-
i sin a)^ = cos ma + i sin ma. This can also be proved for any
real index. The theorem is known as De Moivre's Theorem.
The proof for a positive integer follows at once from 131,
where it is shown that in the product of two complex num-
bers, the modulus of the product is equal to the product of
the moduli of the factors, and the argument of the product
is equal to the sum of the arguments of the factors, two fac-
tors being given. If three factors are given, the product of
two of them, obtained by this principle, may be regarded as
a single factor, and hence the principle for two factors ap-
plies with the same conclusions as before extended to three
factors. And in a similar way we obtain for any number of
factors :

The modulus of the product is equal to the product of the
moduli of the factors.

The argument of the product is equal to the sum of the
arguments of the factors.

Applying these principles to the present case, we have the
result, for the number of factors is m^ the modulus of each
factor is unity, and the argument of each factor a ; hence

MISCELLANEOUS TOPICS 325

the modulus of the product is unity and its argument a+ a
+ • • • + « to m terms = ma. Therefore for a positive integer jn

(cos a + i sin «)"^ = cos ma + i sin ma.

381. Continuity of a Function of a Complex Variable. As
in the case of a function of a real variable 319, so we
define continuity for a function of a complex variable, and
say in particular : A rational integral algebraic function
/(s;), where z= x -{- yi, is continuous at the finite value
z, or at the point (x^ ?/), when it satisfies the condition
|/(2J + A) — /(2) I = 0, as I A I = 0, in whatever direction around
the point (a;, ?/), h may be taken.

It is thus seen that/(2;) = a^z^ + a^z'"''^ + ••• + «« is contin-
uous, for /(2) is single-valued, and the development of 317
holds for/(^ + A), and

/(^ + /0-/(z)=A/'(^) + ^/"(2)+ ... +->"(2).

Z. 71.

As I A 1=0, |/»|.[/'(2)|+l|il -I/" 0)1+ - =0

by 315. A fortiori

I ¥'(^} + |t/"C^) + • • • I = !/(^ + '0 -/(^)1 = 0,

since the modulus of a sum of terms is smaller than or at
most equal to the sum of the moduli of the terms.

382. Geometrical Representation of a Function of a Com-
plex Variable. When f{z) = a^z" -]-a^z'*~'^ -^ ••• + a„ is ex-
panded, it is seen that w =fQz') = w + vi^ where u =f^(x., ?/),

Just as z represents a point (x, ?/), so ^v represents a point
(t*, v^. For distinctness of representation, we shall use two
planes, Figs. 35 and 36, one the z plane or xy plane for repre-

326

COLLEGE ALGEBRA

senting the values of 2, and the other, the w plane or uv
plane for those of w.

From the continuity of w just proved, 381, we see that if
the point P representing z traces a curve in the x^ plane, the

Y
A

0

X

Fig. 35.

point Q representing w traces another curve in the uv plane.
That w =f(z) vanishes is the same as to say that when z is
at P = (x, y\ Qis at 0' = (0, 0).

o'

u

Fig. 36.

383. Isogonality of the Function f{z).

When z starts from P and traces the curves Pa, P5,
Fig. 37, w starts from Q and traces the corresponding curves
Qa' ^ Qh\ Fig. 38, and we shall prove that if f'(z) is neither

MISCELLANEOUS TOPICS

327

Fig. 37

0 nor 00 at ft the angle t\Qt'^ between the tangents to the
curves Qa', Qb', at ft is equal to the angle t^Pt^ between the

Fig. 38.

328 COLLEGE ALGEBRA

corresponding tangents at P, and if /'(^)=/'^(2;)= ••• =
/(->(2;) = 0, /(-+^)(^)^ 0, ^ t\Qt'^ = (m + 1) ^ t^Pt^,

By 128, 130,

OP , Oa , Oa-OP Pa , . , ,

Oa!
similarly w' = w -\-k, where k = ^^^,,

and Oir=Oif' = + l.

Also 1^1 = ]^''^f' f;^= length Pa ; |A:| = length G«^
length C>jM

and as a = P, |7i| = 0, a' = ^, and |^| = 0.

w'=f(z')=fCz + JO,

or if p and r are the moduli and a and (j> the arguments of k
and A respectively,

r(cos9+z sm 9) Zl

Taking the limits of both members as a = P, i.e. as |^| =0,
and observing that (j) = yjr^ « = A we have

\ r J \GOS yjr + z sin sjrj
or rX^ycos(/3-i/r)+zsin(^-'>/r))=/'(2) by 132.

MISCELLANEOUS TOPICS 329

Ph Oh'
Similarly, if h' and k' are -— -, -^, respectively, i/r' and ^'

the corresponding angles to the tangents Pt^, Qt\, r' and p^
the moduli of h' and k',

(i2^^(cos (^' - ^') + isin (/3' - ^/r')) = /'(^).

Hence mod /'(«) = i2, or i^, i?- = ie^,

and if/'(0) is neither 0 nor oo, the arguments /3 — yjr and
/3' — yjr' must be equal, i.e.

^-ylr=/3' -f.or 13'- I3=ylr'-ylr, i.e.
^t',Qt'^ = ^t,Pt,^.

This is called the isogonal property, or isogonality^ of the
function /(s;).

384. Failure of the Isogonal Property.

the isogonal property fails, but is replaced by another angle
property.

In this case k = —^ /(»^+i> (2;) + • • •,

^ 1 /("^+i)(2) + ...,

/(m+l)(^).

Similarly L-f—= ^ f"'^^Kz).

^ h'"'+' (>?i + l)I ^ ^

. /^ jQ A (cos /3 + 2 sin /3) ^fj^ p' \ (coi^^'-hism/3')

330 COLLEGE ALGEBRA

or, by De Moivre's Theorem, and 132,

(^ ;£l)(^^^ (^ - (m + 1) t) + i sin (^ - (w + 1) ./r)^

= (-^ /Cri)(^^<^' - (m + 1) ^/^O + ^ sin(/3^ - (m + 1) ^/r')^
=/('"+!> (2).

.*. L-^—; = L ; . and since f^^+^Y^^ is neither 0 nor 00, the

arguments /3 — (m + 1) i/r and y8' — (m + 1) t/t' are equal,

whence /3' - y8 = (m + 1) (i/r' - i/r),

or, ^ f \ ^^'2 = (^ + 1) ^ hPt^.

385. The Proof that Every Algebraic Polynomial a(^"-{-a^z"~'^

-\ f-fl/j has a Root. We prove this proposition, known as

tlie fundamental proposition of algebra, by indirect reasoning.
If possible, let tliere be no value of z for which f(z^ = 0.
Then it follows that there must be a value of /(^), repre-
sented by Q and corresponding to a value of z^ represented
by P, and having a modulus, such tliat for no other value of
f(z) is the modulus smaller ; that is for no other value of
f(z) can the point representing /(2) be nearer the origin 0' ,
We consider two cases :

(1) /(^) ^ 0,

(2) f\z~) =f'(z') = ... =f^^-\z') = 0, /(-+i)(^) ^ 0.

1. Since for the point Q, f'(z) is neither 0 nor 00, and
since f(z) represented by Q is continuous at §, if z moves
5 out in all posible directions on rays from P once around,
Fig. 39, IV by virtue of its continuity and isogonality, 381,
383, will move out in all possible directions from Q once
around. Fig. 40, and therefore will somewhere move nearer

MISCELLANEOUS TOPICS

331

to the origin than Q itself. Hence the conclusion that there
is a value of f(z) having a modulus such that for no other
value of /(2!) is the modulus smaller, is false, and hence also

I

0

X

Fig. 39.

the premise that there is no value of z for which f(z) = 0,
from which this conclusion followed. Therefore for some
value of 2, «q2^ + a^a;""^ + ••• + a;j vanishes.

V

O'

u-

Fig. 40.

2. In this case we cause z to move out from P, until the

rays from P cover a sector of the way around, then

m + 1

by 384, to will have moved out from Q in all possible direc-
tions once around, and the result is the same as in case 1.
Hence a^z^^ + a-^z^~'^-\- ••• -\- a^ always has a root.

INDEX

(The numbers refer to pages)

Abscissa, 2.

Algebraic polynomial, root of, 330.

Amplitude of complex number, 85, 88.

Antecedent, 20.

Argument of complex number, 88.

Arithmetical progression, 95.

Auxiliary series, 157.

Axes, 2.

Base of a system of logarithms, 217.
Binomial coefficients, 126, 130.

series, convergency of, 171.
surd, extraction of square root of, 73.
theorem, any real exponent, 169, 321.
general term, 129.
greatest term, 133.
positive integral exponent, 124.
Binomial quadratic surds enter equations in pairs, 268.
Biquadratic equation, 290.

Characteristic of a logarithm, 222.
Circle, coordinates of center, 8.
equation of, 6, 7.
radius of, 8.
Coefficients, binomial, 126, 130.
Cologarithms, 227.
Combinations, 113.

complementary, 116.
Commensurable numbers, 50.
Complex number, 52, 79.

amplitude of a, 85, 88.

argument of a, 88.

conjugate of a. 80.

graphical representation of a, 82, 85.

modulus of, 80.
Complex numbers, identity theorems for, 81.

333

334 INDEX

Complex roots enter equations in pairs, 268.

Complex variable as function of its modulus and argument, 88, 323.

continuity of, 325.
Compound ratio, 20.
Conjugate complex numbers, 80.
Conjugate trinomial surds, 71.
Consequent, 20.
Constants, 139.
Continued fractions, 185.

convergents of, 187.

alternately greater and less, 194.
closer and looser limits of error of, 195.
lowest terms, in their, 194.
recurring, 188.

root of quadratic equation, 188.
Continuity of a function of a complex variable, 325.
Convergency and divergency of series, 147.
Convergency of binomial series, 171.

geometrical series, 150.
some particular series, 312.
Coordinates, 2.
Cube roots of unity, 271.
Cubic equation, solution of, 287.
Cubic, reducing, 291.
Curve, to plot, 4.

Decomposition of fractions, 173.
De Moivre's theorem, 324.
Derived functions, 264.

geometrical interpretation of, 280.
Descartes' rule of signs, 267.
Determinants, 244*.

column of, 245.
elements of, 246.

conjugate, 246.
self-conjugate, 246.
minors of , 253.

first, 254.
principal diagonal of, 246.
product of two, 259.
row of, 245.

secondary diagonal of, 246.
solution of linear equations by, 256.
terms of, 246.

principal or leading, 246.

INDEX 335

Development of a fraction into a series, 1G6.

general term in, 182.
Development of a function, 263.
Differences, finite, 206.

orders of, 207.
Discriminant of cubic, 289.

of quadratic, 38.
Distance between two points, 2, 8.
Duplicate ratio, 21.

Equal roots of equations, 36, 289, 294.
Equation of locus, 2.
Equations of first degree, 6.

graph of, 4.

represent straight lines, 6.
higher degree, 6.

graphs of, 6.
quadratic, theory of, 32.
radical, 74.

extraneous roots of, 77.
solutions by determinants, 256.
theory of, 262.
Error, closer and looser limits of, 195.
Euler's cubic, 291.
Exponential function, 231.
theorem, 241.
Exponentiation, 217.
Exponents, theory of, 52.
Extraneous roots of radical equations, 77.

Factorial 7i, 111.

Factoring of quadratic expressions, 34.

symmetric and related expressions, 272.
Factor theorem, 44.
Finite differences, 206.

orders of, 207.
Fractions, continued, 185.

convergents of, 187.
infinite, 185.
recurring, 188.

root of quadratic equation, 188.
terminating, 185.
decomposition of, 173.
development of, into series, 166.
partial, 173.

336 INDEX

Function, derived, 264.

development of, 263.

exponential, 231.

generating, 205.

graphic representation of, 1.

isogonality of, 326.

failure of, 329.

of a complex variable, geometrical representation of, 325.

symmetric, 271.
Fundamental proposition of algebra, 330, 331.

General term in the development of a fraction, 182.
Generating function of a recurring series, 205.
•Geometric addition, 86, 89.
division, 91.
multiplication, 86, 90.
subtraction, 89.
Geometric interpretation of the derived function, 280.
Geometrical progression, 99.

convergency of, 150.
Graph, 4.

of equation of first degree, 4.

higher degree, 6.
Graphic representation of a function, 1.

a point, 1.

complex numbers, 82, 85.
direct variation, 28.
inverse variation, 29.
real and complex roots, 39, 40.
Graphical solution of simultaneous equations, 11.

Harmonical progression, 103.
Harmonic division, 104.

series, 158.
Horner's method, 297.

Identity of two polynomials, 46.
theorem, 46.

theorems for complex numbers, 81.
Imaginary numbers, 51.
Indefinitely great numbers, 305.
small numbers, 305.
Indeterminate equations of the first degree, 196.

general solutions of, 197.
particular solutions of, 197.

INDEX 337

Indeterminate forms, 142, 807.
Indices, tlieory of, 52,
Induction, matliematical, 125, 302.
Inequality, 13.

definitions, 13.

fundamental, 17.

in same or opposite sense, 14.

members of, 13.

notation, 13.
Infinites, 305.
Infinitesimals, 305.
Interpolation formula, 213.
Inverse ratio, 21.

Involution and evolution of surds, 67.
Irrational numbers, 58.
Isogonality, 325.

Limits, 139, 305.
Linear equation, 0.
Locus, 4.

equation of, 3.
symmetrical, 6.
Logarithmic series, 241.
Logarithms, 217.

Briggs, 221.
calculation of, 242.
characteristic of, 222.
common, 221.
mantissa of, 222.
Napierian, 221.
natural system of, 221.

base of, 237.
numbers corresponding to, found from table, 226.
of numbers found from table, 223.
table of, 224, 225.

use of, 223.

tabular difference of, 226.

Mantissa of a logarithm, 222.
Mathematical induction, 125, 302.
Minors of determinants, 253.
Modular series, 161.
Modulus of a complex number, 80.
Multinomial theorem, 134.

fireneral term of, 134.

338 INDEX

Napierian base, 237.
Natural system of logarithms, 221.
Number of real roots of an equation, 292.
Numbers, commensurable, 50.

complex, 52, 79.

incommensurable, 51.

rational and irrational, 58.

real and pure imaginary, 51.
Numerical equations, 296.

Ordinate, 2.

Origin, 2.

Oscillating series, 150, 151.

Partial fractions, 173.
Path, equation of, 3.
Permutations, 108, 109.
Plotting a point or curve, 4.

a straight line, 6.
Point, coordinates of, 2.

representation of, 1.
Points of intersection of two curves, 7.
Polynomials, identity of two, 46.
Product of two infinite series, 320.
Progression, arithmetical, 95.

common difference of, 95.
means, 97.
geometrical, 99.

constant ratio of, 99.
means, 101.
harmonical, 103.

means, 103.
Proportion, 22.

by addition, 23.
by alternation, 22.
by composition and division, 24.
by inversion, 23.
by subtraction, 24.
continued, 24,
extremes and means of, 22.
Proportional, mean, 25.
third, 25.

INDEX 339

Quadrants, 2.

Quadratic equations, discriminant of, 38.

formation of, with given roots, 33.
roots of, 32.

graphical representation of real and complex,

39, 40.
nature of, 36.
two and only two, 41.
zero and infinite, 36, 37.
solution of, by inspection, 35.
theory of, 32.
Quadratic expressions, factoring of, 34.
surds, properties of, 72.

Radical, entire, 58.

equations, 74.
mixed, 58.
Radicals, 57.
Ratio, 20.

antecedent of, 20.
compound, 20.
consequent of, 20.
duplicate, 21.
inverse, 21.
of greater inequality, 20.

lesser inequality, 20.
subduplicate, 21.
sub triplicate, 21.
terms of, 20.
triplicate, 21.
unit, 20.
Rational number, 68.
Real numbers, 51.
Recurring series, 203.

generating function of, 205.
order of, 203.
scale of relation of, 203.
sum of n terms of, 204.
Reducing cubic of biquadratic, 291.

quadratic of cubic, 287.
Remainder after n terms in a series, 148.

theorem, 48.
Representation of a point, 1.

function, 1, 325.
Rolle's theorem, 282.

340 INDEX

Root, square, 50.

every equation of the nth degree has a, 330.
Roots, every equation of tlie ?ith degree has w, 45.
extraneous, 77.
of a quadratic equation, 32, 36.

two and only tw^o of, 41.
zero and infinite of, 36, 37.

Scale of relation, 203.
Series, 147.

absolutely convergent, 147, 161.

auxiliary, 157.

convergency of, 148-163, 171, 312, 320.

convergent, 147.

divergent, 147.

exponential, 241.

finite, 147.

harmonic, 158.

infinite, 147. -

the product of two, 320.

logarithmic, 241.

modular, 161, 317.

necessary and sufficient conditions for convergency of, 148, 149.

oscillating, 150, 151.

recurring, 203.

scale of relation of, 203.

remainder of, after n terms, 148.

summation of, 183, 199, 303.

tests for convergency of, 151.
Simultaneous equations, 7.

geometrical representation of, 7.
solution of, by graphical methods, 10, 11, 12.
Solution of biquadratic equation, 290.
cubic equation, 287.
quadratic equations by inspection, 35.
Sturm's theorem, 292.

functions, 292, 298.
Summation of series, 183, 199, 303.
Surds, 57.

addition of, 62.

comparison of, 67.

division of, 66.

entire, 58.

involution and evolution of, 67.

mixed, 58.

INDEX 341

multiplication of, 63.

quadratic, properties of, 72.

square root of binomial, 73.

rationalization of, 66-68.

similar, 61.

subtraction of, 62.
Symmetrical locus, 6, 7.
Symmetric functions, 271.

Table of logarithms, 224, 225.
Tabular difference, 226.
Theory of equations, 262.

indices, 52.

quadratics, 32.
Total degree, 158, 313.

Undetermined coefficients, 164.

Vandermonde's theorem, 321.
Variables, 139.
Variation, 28.

direct, 28.

inverse, 28, 29.

joint, 28, 29.

{/

^/'T

!*•

14 DAY USE

RETURN TO DESK FROM WHICH BORROWED

LOAN DEPT.

This book is due on the last date stamped below, or

on the date to which renewed.

Renewed books are subject to immediate recall.

^^|'66l IRCp

NOV 6 - 1966

^-^

•DECEIVED

NOI/ 5 '66.

'zm

LOAN DEF»T

r

m

LD 21A-60»i-10,'65
(P7763sl0)476B

General Library

University of California

Berkeley

IVI306CI57

THE UNIVERSITY OF CALIFORNIA LIBRARY

4

* ■**«%

-'3fl* vi ;-. » V^ ' *